limits

Rules to the Limits

 fraction with two polynomials divided, use the rules:
1. If the highest exponent of the top is equal to the highest exponent of the bottom, then you put the leading coeff over the leading coeff.

2. If the highest exponent of the top is greater than the highest exponent of the bottom, then you use plus or minus infinity (plug in to see if you get positive or negative)

3. If the highest exponent of the top is less than the highest exponent of the bottom, then it is equal to zero.

**If it doesn't follow the rules,
1. Plug into y=

2. Use the 2nd funtion, then table on your calculator.
3. Plug in 10, 100, 1000, 10000, 100000....

(In the table, if e is negative, then it is equal to zero. If e is positve, then it is equal to infinity.)

Until you see a pattern.

**If it is a geometric(a number raised to n) and |r| <1, then limit=0
If it is greater than 1, then it is infinity.

Example 1:

lim (n^2 -n)/n^3
= 0

lim [3n^2/n]
= infinity

lim (7n^2+5n)/(10n^2)
= 7/10

Limits

This past week of school we were introduced to Calculus. Our first lesson was in Chapter 19, on Limits. The lesson was pretty simple because we did lessons on limits in advanced math.

Notes to take:

A limit is a singular y-value.

-To find a limit, plug into the equation. If you have 0 on the bottom and 0/0, you have to try doing something else.

Other options:

1-Graph (infinite/-infinite is still considered a (y) value.

2-Use the table:

x-.1, x-.01, x-.001, x, x + .001, x + .01, x + .1

à              ß                              

   

3-Trying to factor and cancel something (algebra).

Examples:

11)   Lim x->4

x-4 / x^2-3x-4

x         3.9     3.99    3.999  4.001   4.01    4.1

f(x)   .209   .2009  .2009   1.9996 1.996  1.96

From doing the table, you can see that as the table values approach x, it gets closer to .2. Therefore, .2 is the answer.

Analytic geometry

Definition of Circle

Definition: A circle is the set of all points that are the same distance, r, from a fixed point.

General Formula: X ² + Y ²=r² where r is the radius

• Unlike parabalas, circles ALWAYS have X ² and Y ² terms.
• X² + Y²=4 is a circle with a radius of 2 ( since 4 =2²)
• Remember that a circle is a locus of points. A circle is all of the points that are a fixed distance, known as the radius, from a given point, known as the center of the circle.

Find the equation of a circle having endpoints (3,7) and (5,-2).

center:  (3-5/2, 7-2/2)=(-1,5/2)

D:  square root of (-5-3)^2+(-2-7)^2

=square root of 64+81

=square root of 145

=12.04

=12.04/2=6.02

(x-h)^2+(y-k)^2=r^2

(x+1)^2+(y-5/2)^2=36.24

Review! :)

What is the formula for an arithmetic sequence?
****tn=t1+(n-1)d

Find the formula for the sequence 10,15,20,25,30,.....
You use the tn=t1+(n-1)d
Tn=10+(n-1)5
Tn=10+ 5n-5
****Tn=5n+5


how many positive two digit numbers are there that are multiples of three?
12,15,.....99
Plug it into your tn=t1+(n-1)d
99=12+(n-1)3
99=12+3n-3
99=9+3n
90=3n
****N=30

what is the geometric sum of the series formula?
***sn=t1(1-r^n)/1-r


Find the slope of the parallel line to: 4x-y=6.
Plug it into -a/b.
-4/-1
****4

Find the slope of the perpendicular line to:3x+2y=4.
-a/b
-3/2
***2/3

Hyperbolas & Identifying graphs

*Standard form
x^2/a^2-y^2/b^2=1

1. the major axis is the variable with the largest denominator (positive)
2. the minor is the variable with the smallest denominator (negative)
3. The vertex is the square root of the largest denominator (put in point form)
4. The other value is the square root of the smallest denominator (put in point form)
5. Skip
6. Skip
7. Focus - F^2=largest denominator + smallest denominator
8. Horizontal Asymptotes - x= + or - square root of y denominator / square root of x denominator 
9. Graph

Ax^2+Bxy+Cy^2+Dx+Ey+F=o *standard form*

-To find the shape of the graph without using the standard form, plug into B^2-4AC

• If it is a circle, you get a negative number,  A=C & B=o
• If it is an ellipse, you get a negative number, A doesn't = C, *& B doesn't = 0
• If it is a parabola you get 0
• If it is a hyperbola, you got a positive number

Example 1: Identify the graph of the equation : X^2-2xy+3y^2-1=0

A=1   B=-2   C=0

B^2-4AC

(-2)^2-4(1)(3)

4-12=-8

Ellipse

1) Take the coefficients of your A, B, & C term

2) Plug them into your formula

3) Solve, then look back at your notes when you find your answer to identify what the graph is

Review of sums of infinite series

This week we haven’t been doing much but reviewing lessons from the beginning of school since finals are approaching. So Im going to do this blog on reviewing lesson 13-5, Sums of infinite series since this lesson is in our review packet!

Notes to know:

-Only geometrics where lrl < 1 have an infinite sum.

The sum formula for geometric is S= t1/1-r

-To find where an infinite geometric converges, set lrl < 1 and solve for x.

-To write a repeating decimal as a fraction, do this:

Whats repeating/place – 1

Examples:

1 1)      Find the sum of the infinite geometric series.

9-6+4-…

To find your R: -6/9= -2/3 , 4/-6= -2/3 (r=-2/3)

S=9/1 – (-2/3)= 9/1+2/3= 27/5

2 2)      For what values does the series converge?

1+ (x-2) + (x-2)^2 + (x-2)^3

l x-2 l < 1

-1 < x-2 < 1

Add 2 to each side and your answer is 1<x<3

3 3)      Repeating decimals: Write .454545.. as a rational number

45/100-1 = 45/99= 5/11

Arithmetic & Geometric Sequences!

*Sequence-a list of numbers. Two types of sequences: --Arithmetic sequence-made by adding the same positive or negative interger to generate the sequence -Fromula tn=t1=(n-1) d -t1=first term -n=term # -d=difference (what is being added or subtracted) -tn=last term --Geometric sequence-made by mulitiplying by the same whole number or fraction to generate a sequence. -Fromula tn=t1 * r ^(n-1) -r=ratio the number that is mulitiplied to generate the sequence -t1=first term -n=term # (tn= ___ term) There are many ways to generate a sequence using these formulas or to find the formula for a sequence of numbers. _____________________________________________________ Ex.1: Find a formula for tn. 1, 4, 7, 10, ... d=3 t1=1 n=n tn=t1+(n-1)d 1+(n-1)3 =1+3n-3 tn=3n-4 Ex.2: Is the following sequence arithmitic or geometric? Find the formula for the nth term. 3,5,7,9,11... This would be arithmitic because we are adding 2. Therefore d = 2. To find the formula for the nth term we need to plug numbers into the formula: tn = t1 + (n-1)d tn = 3 + (n-1)2 tn = 3 + 2n - 2 tn=2n+1

Combination and Permutation

Permutation means when the order is importance
The formula used is : nPr=n!/(n-r)!
When order isnt important you use combination  
The formula used is:Cr=n!/(n-r)!r!


Example:what order could 16 pool balls be in?After choosing, number "14" you can't choose it again.

Ex 2:Permutation: 16 × 15 × 14 × 13 × ... = 20,922,789,888,000

Combination:Picking a team of 3 people from a group of 10. 


C(10,3) = 10!/(7! * 3!) = 10 * 9 * 8 / (3 * 2 * 1) = 120.


Ex 3:Permutation: Picking a President, VP and Waterboy from a group of 10. 
P(10,3) = 10!/7! = 10 * 9 * 8 = 720.


Combination: Choosing 3 desserts from a menu of 10. 
C(10,3) = 120.


Permutation: Listing your 3 favorite desserts, in order, from a menu of 10.


 P(10,3) = 720.


Ex 4:Given 4 people, Bob, Mike, Sue and Alice, how many different ways can these three people be arranged where order matters?

Finding all possible arrangements of Bob, Mike, Sue and Alice where order matters would be:
BMSA,    BMAS,    BSMA,    BSAM,    BAMS,    BASM
MBSA,    MBAS,    MABS,    MASB,    MSBA,    MSAB
SBMA,    SBAM,    SMBA,    SMAB,    SABM,    SAMB
ABMS,    ABSM,    AMBS,    AMSB,    ASBM,    ASMB

There are 24 ways to arrange the four people four at a time,   or   4!

A series is a list of numbers being added. A sequence is a list of numbers. An arithmetic sequence is a sequence that is generated by adding the same number each time. Arithmetic Sequence formula tn=t1 + (n-1)d t1= first term n= term number d= difference  tn= ___ term A geometric sequence is a sequence that is generated by multiplying the same number each time. (to divide, we use fractions) formula: tn=t1(r)^(n-1) t1= first term r= what is being multiplied n= term number tn= ___ term Example: 1.) Find the first four terms of the sequence and state whether the sequence is arithmetic, geometric, or neither. tn= 4n+12 n=1 4(1)+12=16 n=2 4(2)+12=20 n=3 4(3)+12=24 n=4 4(4)+12=28 It is arithmetic because you are adding 4 each time! Example: 2.) Find the first four terms of the sequence and state whether the sequence is arithmetic, geometric, or neither. tn=-2n+8 n=1 -2(1)+8=6 n=2 -2(2)+8=4 n=3 -2(3)+8=2 n=4 -2(4)+8=0 It is arithmetic because you are adding -2 each time

Vectors in Three Dimensions!

Helloo,
I hope everyone had a wonderful weekend!

distance: square root of (x2-x1)^2  +(y2-y1)^2 + (z2-z1)^2
midpoint: ( x1+x2/2 , y1+y2/2 , z1+z2/2 )
sphere: (x-x0)^2 + (y-y0)^2 + (z-z0)^2 = r^2
vector equation: ( x1 y1 z1 ) = ( x0 , y0 , z0 ) + t( a , b , c )
parametric: x=x0+at  y=y0+bt  z=z0+ct
lml (magnitude): square root of x^2 + y^2 + z^2
u * v: x1 x2 + y1 y2 + z1 z2

Example 1:
Find the length and midpoint of AB.
A = ( 2 , 5 , -3 ) and B = ( 0 , 3 , 1 )
Midpoint: ( 2 + 0 / 2 , 5 + 3 / 2 , -3 + 1 / 2 ) = ( 1 , -2 , 1 )
Distance: square root of ( 0 - 2 ) ^ 2 + ( 3 - 5 ) ^ 2 + ( 1 + 3 ) ^ 2 = square root of 4 + 4 + 16 = square root of 24 = 2 square root of 6

Example 2:
Find the length and midpoint of AB.
A = ( 3 , -5 , 0 ) and B = ( -1 , 1 , 2 )
Midpoint: ( 3 + 1 / 2 , -5 + 1 / 2 , 0 + 2 / 2 ) = ( 1 , -2 , 1 )
Distance: square root of ( -1 - 3 ) ^ 2 + ( 1 + 5 ) ^ 2 + ( 2 + 0 ) ^ 2 = square root of 16 + 36 + 4 = square root of 56 = 2 square root of 14

Example 3:
Simplify the expression.
a. ( 3 , 8 , -2 ) + 2 ( 4 , -1 , 2 )
( 3 , 8 , -2 ) + ( 8 , -2 , 4 ) = ( 11 , 6 , 2 )
b. ( 1 , -8 , 6 ) * ( 5 , 2 , 1 )
5 + -16 + 6 = -5
c. magnitude of ( 3 , 5 , 1 )
square root of 3^2 + 5^2 + 1^2
= square root of 9 + 25 + 1
= square root of 35

Combination and Permutation

Permutation means when the order is important. You use this formula: nPr=n!/(n-r)! Combination means when the order is not important. You use this formula:Cr=n!/(n-r)!r! Example 1: 5people are standing in a line.  How many different outcomes are possible?   P(5, 5) = 5! = 5 × 4 × 3 × 2 × 1 = 120 Example 2:  In how many ways can you select a committee of 4 people from a group of 10 members? C(10, 4) = 210 Example 3: License plates for cars have to be unique. If a license plate contains 6 characters consisting of 2 letters followed by 4 digits  example:  QW2354,  A-how many different license are possible? B-how many different license are possible if letters were allowed to repeated but numbers are not allowed to be repeated? There are 26 × 26 × 10 × 9 × 8 × 7 =  3,407,040 different plates with no repeating numbers. Example 4: In how many ways can you select a committee of 3 people from a group of 12 members? C(12, 3) = 220

Function Notation

f(x) = x^2 + 5x - 2 is an example of a function. If it says "Evaluate f(2)", you can assume x = 2 and plug in 2 for all x's in the function and simplify. Here are some other things you can do with functions:

(f+g)(x) = f(x) + g(x)
(f-g)(x) = f(x) - g(x)
(f*g)(x) = f(x) * g(x)
(f/g)(x) = f(x) / g(x)
(f○g)(x) = f(g(x)) or plug in the equation g(x) into all x's in f(x)
(g○f)(x) = g(f(x)) or plug in the equation f(x) into all x's in g(x)


Ex. 1

f(x) = 3x+3
g(x) = x-8

Evaluate f(2)
f(2) = 3(2)+3
= 6+3
= 9

Evaluate g(3)
g(3) = (3)-8
= -5


Ex. 2
f(x) = 2x-8
g(x) = x^2 +4

Find (f+g)(x)
2x-8 + (x^2 + 4)
x^2 + 2x -4

Find (f-g)(x)
2x-8 - (x^2 + 4)
2x - 8 - x^2 - 4
-x^2 + 2x - 12

Find (f*g)(x)
(2x-8)(x^2+4)
2x^3+8x-8x^2-32
2x^3-8x^2+8x-32

Find (f/g)(x)
(2x-8)/(x^2+4)

Find (f○g)(x)
2(x^2+4) - 8
2x^2+8-8
2x^2

Find (g○f)(x)
(2x-8)^2 +4
4x^2-32x+64+4
4x^2-32x+68


Ex. 3
f(x) = 2x^2 -1
g(x) = x+1

Find (f○g)(2)
2(2+1)^2 -1
2(3)^2 - 1
2(9) - 1
18 - 1
17

Find (g○f)(2)
(2(2)^2 -1) +1
2(4) -1 +1
8 -1 +1
8

Find (g*f)(1)
(1+1)(2(1)^2-1)
(2)(2-1)
(2)(1)
2

Find (f+g)(8)
2(8)^2 - 1 + 8 + 1
2(64)-1+8+1
128+8
136

Find (f-g)(6)
2(6)^2 -1 -(6+1)
2(36) - 1 - 7
72 -8
64

Ch. 6

Find the equation of a circle having endpoints (3,7) and (5,-2).
center:  (3-5/2, 7-2/2)=(-1,5/2)
D:  square root of (-5-3)^2+(-2-7)^2
=square root of 64+81
=square root of 145
=12.04
=12.04/2=6.02
(x-h)^2+(y-k)^2=r^2
(x+1)^2+(y-5/2)^2=36.24

In order to find the equation for a circle, you have to find the center and the radius.  To do this, we use the midpoint formula, and the distance formula.  The midpoint formula is (x1+x2/2,y1+y2/2).  We are given tow different points, so we use those two points and plug them into to formula I just gave you.  Once you do that, you get (-1,5/2).  Now you have to use the distance formula to find the radius.  The distance formula is the square root of (x2-x1)^2+(y2-y1)^2.  You again use the two points you were given in the problem to solve for the distance.  Once you solve this, you get 12.04.  12.04 is the diameter, and we need the radius to find the equation of a circle, so all you do now is divide 12.04 by 2 and we have 6.02 now.  Now we just plug the answers we found into the circle formula.  The circle formula is (x-h)^2+(y-k)^2=r^2.  You have to square your radius you found and that will give you 36.24.  So your answer is (x+1)^2+(y-5/2)^2=36.24.

limits

**If you have a fraction with two polynomials divided, use the rules:
1. If the highest exponent of the top is equal to the highest exponent of the bottom, then you put the leading coeff over the leading coeff.

2. If the highest exponent of the top is greater than the highest exponent of the bottom, then you use plus or minus infinity (plug in to see if you get positive or negative)

3. If the highest exponent of the top is less than the highest exponent of the bottom, then it is equal to zero.

**If it doesn't follow the rules,
1. Plug into y=

2. Use the 2nd funtion, then table on your calculator.
3. Plug in 10, 100, 1000, 10000, 100000....

(In the table, if e is negative, then it is equal to zero. If e is positve, then it is equal to infinity.)

Until you see a pattern.

**If it is a geometric(a number raised to n) and |r| <1, then limit=0
If it is greater than 1, then it is infinity.

Example 1:
Find the given limit.
lim (n^2 -n)/n^3
= 0

lim [3n^2/n]
= infinity

lim (7n^2+5n)/(10n^2)
= 7/10

Area of a non-right triangle

To find the area of a non-right triangle, use this formula:
Area=1/2(leg)(other leg)sin(angle between)
So,

EXAMPLE
________________________________________

    78 degrees
    /\
9 /  \  12
  /__\              
                         
In triangle ABC, a = 12, B = 78 degrees, and c = 9. Find area of the triangle.
Area=1/2(12)(9)sin(78)
Area=15.82

Chapter 4

Domain and Range:-Domain- the interval of x values where the graph exist
-Range- the interval of y values whre the graph exist
-Zeros, x-intercept, root - set = 0, slove for x
-To be a function the graph must pass the vertical line test
-If given points the domain is the list of all x-values and range is of all y-values in { }

To find Domain and Range:
-Polynomials: domain and always (negative infinity, positive infinity)
-Square root: 1. Set inside root equal to 0 and solve
                      2. Set up intervals
                      3. Plug in if you get a negative x interval
                      4. Write answer in interval notation- use brackets by the number Range: [ a, infinity)
-Square root: Domain: 1. Set inside equal to 0 and solve
                                     2. Put answer in [-   ]
                      Range: [0, square root of number] if a=0
                                  [0 plus or minus a, square root of a number + a] if a is positive
                                  [0 minus a, square root of a number minus a] if a is neagtive
-Fraction: Domain: 1. Factor top and bottom
                                2. Cancell if possible and marke # if cancelled
                                3. Set bottom equal to 0 and slove for x value
                                4. Write in interval notation stopping at numbers found in 2 and 3
-Absolute Value: Domina: (- infinity, positive infinity)
                             Range: [0 + a, infinity) if v is ponted up and a is positive
                                         [0 - a, infinity) if v is pointed up and a is negative
                                         (- infinity, 0 - a] if v is ponted down and a is negative
                                         (- infinty, 0 +a] uf v is pointed down and a is positive


Example 1: Find the Domain and Range of the following: f(x) = 5x^3 + 4x^2 + 3x - 6

 Domain: (- infinity, infinity)
 Range: (- infinity, infinity)

Funtion? Yes


--Daniellee

Circles

Standard form (x – h)^2 + (y – k )^2 = r^2

Where (h,k) is the center and r is the radius

You can find the radius using the distance formula with the center

and a point on the outside or you can half the diameter found by

using 2 points on the outside through the center

Equations must be put in standard form by completing the square

a. Divide by leading coefficient, move all numbers to the right

b. Half the middle term, then square it and add to both sides

c. Factor

To graph in your calculator you must solve for y and put +

and – in formula in y =

To find the intersection of a line and circle solve the line for y

Plug into the circle equation for y

Solve the quadratic

Ex 1. Find the center and radius of (x – 3)^2 + (y + 7)^2 = 19

Center: (3, 7) Radius: sqrt19

Ex 2. Given C(-1, 7) and the radius is 7 units, find the equation

of the circle in standard form.

(x + 1)^2 + (y – 7)^2 = 49

Ex 3. Find the center and radius of x^2 – 2x + y^2 – 6y = 9

-2/2 = (-1)^2 = 1 -6/2 = (-3)^2 = 9 9 + 1 + 9 = 19

(x – 1)^2 + (y – 3)^2 = 19

Center: (1, 3) Radius: sqrt19

Conics

Conics are a type of graph. They can be graphed by calculator or hand.  Each conic has a certain equation and a way it looks when graphed. This is about circles.

Standard form: (x-h)^2 = (y-k)^2 = r^2

-You can find the radius using the distance formula with the center and a point on the outside or you can half the diameter foud by using 2 points on the outside through the center.

-Equations must be put in standard forn by completing the square. These are the steps:
1.Divide by leading coefficient, move all #s to right
2.Half the middle term, them saure it and add to both sides
3.Factor everything

-To graph in your calculator you must solve for y and put a + e - formula in the Y= slot in calculator.

-To graph without calculor you  must find center points and raidus. Graph as told.

EX1.  Write an equation for the line with a center of (2,8) and radius of 4.
plug into formula (x-h)^2 + (y-k)^2 = r^2
(x-2)^2 + (y-8)^2 = 16

EX 2. Find the center and radius of (x-3)^2 + (y +5)^2 = 9
center = (3, -5)
radius + sqaure root 9 = 3

Circle Stuff

The equation of a circle and the properties of the circle. The equation used is the standard equation that has the form: (x - h)2 + (y - k)2 = r2 where h and k are the x- and y-coordinates of the center of the circle and r is the radius. The exploration is carried out by changing the parameters h, k and r included in the above 1. h and k to zero and parameter r to 1. Check that the circle shown has the center at (0,0) and radius equal to 1. 3. set r to zero and parameters h and k to different values, the graph of the circle is a point, Explain.(Hint:Solve the equation (x - h)2 + (y - k)2 = 0 4 - Keep r equal to 1 and shift the circle by changing h and k. Check that the center of the circle is at (h , k). 5 - Keep h and k constant and change r. Check that the circle has radius r. 6 - Set h, k and r to 1. The circle has one point of intersection with the x-axis and one point of intersection with the y-axis. These are called the x and y intercepts. Find these points analytically using the equation of the circle. (x - h)2 + (y - k)2 = r2 (Hint: To find the x-intercepts set y = 0 in the equation and solve for x. To find the y-intercepts set x = 0 in the equation and solve for y.) 7- Set r to 2 and h to a certain value. Change k from -1.8 to 1.8 (|h| less than r). How many x-intercepts are there? Set k to 2 (the radius), How many x-intercepts are there? Set k to -2, how many x-intercepts are there? Set k to values greater than 2 (the radius), how many x-intercepts are there? Set k to values smaller than -2, how many x-intercepts are there? Explain analytically. 8- Try the same exploration as in 7 above with the y-intercepts by changing the value of h. 9- Exercise: Find (analytically) values of h, k and r such that the circle associated with these values has no x or y-intercepts. Check your answer graphically. From the website: http://www.analyzemath.com/CircleEq/CircleEq.html

VECTORS

This past week in advanced math we learned about Dot product.
This lesson was pretty simple. 


Dot product= u*v = <x1,y1> * <x2,y2> = x1 x2 + y1 y2


Here are the rules to know when approaching dot product equations:


-If the dot product equals 0, then the vectors are orthogonal (perpendicular).


-If the vectors are multiples of each other, then they are parallel.


-In order to find the angle b/w two vectors use: cos theta = u*v over the magnitude of u and v




Example 1: u(2, -1) v(3,6) w(-5,3)


Find the u*v and v*w. Show that u and v are orthogonal and v and w are parallel


u*v = 2(3) + -1(6) = 0 which means they are perpendicular because it equals 0


v*w = 3(-5) + 6(3) = 3


w/v = 18/-15 = -6/5
-They are multiples of each other therefore they are parallel.


Example 2: To the nearest degree find the angle between the vectors (1,2) and (-3,1)


u*v = 1(-3) + 2(1) = -1


magnitude of u = square root of 5
magnitude of v = the square root of 10


-Now following the formula in point 3 we need to do cos theta
1/ square root of 5 times square root of 10
theata = inverse of cos(-1/ square root of 50) = 81.867
-You find it in the 1st and 3rd quadrents: 98 and 261


wink ;-)

Doing this right before 12>

Dot Product has many types according to the 4 below:
u * v = <x1, y1> * <x2, y2> = x1x2 + y1y2
If it equals 0, then the vectors are perpendicular.
If the vectors are multiples of each other then they're parallel.
Properties of Dot Products:
1. Commutative: u * v = v * u
2. Squared: u* u =! u^2; u * u = |u^2|
3. k(u * v) = ku * v
4. u * u = |u|^2

Ex: u(3, -6) v(4,2) w(-12,-6) -Find the u*v and v*w. Show that u and v are orthogonal and v and w are parallel u*v = 3(4) + -6(2) = 0 which means they are perpendicular because it equals 0 v*w = -12(2) + 2(-6) = -60 w/v = -12/4 = -3 -6/2 = -3 : They are multiples of each other therefore they are parallel

DOT Product: u*v= * =x1x2+y1y2 If the dot product equals zero, then the vectors are orthogonal, which means perpendicular. If the vectors are multiples of each other, then they are parallel. To find the angle between two vectors, use the formula: cos(theta)= u*v/magnitudeU * magnitudeV  Properties of the DOT product: 1.  commutative: u * v = v * u 2.  squared u * u = magnitude of u^2 3.  K(u * v)= Ku * v doesn't distribute 4.  u * (v+w) = u * v + u * w Example 1: Find (2,3) * (4,-5) 2(4)+3(-5) =-7 Example 2: Find (3/5,4/5) * (1/2,-3/2) 3/5(1/2)+4/5(-3/2) =-9/10 Example 3: Find the value of a if the vectors (6,-8) and (4,a) are parallel. a/4=3/6 6a=12 divide both sides by 6 a=2  Example 4:  u(3, -6) v(4,2) w(-12,-6) -Find the u*v and v*w. Show that u and v are orthogonal and v and w are parallel.   u*v = 3(4) + -6(2) = 0 which means they are perpendicular because it equals 0  v*w = -12(2) + 2(-6) = -60 w/v = -12/4 = -3 -6/2 = -3 : They are multiples of each other therefore they are parallel

Dot Product

DOT Product: u*v= * = x1x2+y1y2 -- If the dot product equals zero, then the vectors are orthogonal, which means perpendicular. -- If the vectors are multiples of each other, then they are parallel. -- To find the angle between two vectors, use the formula: cos(theta)= u*v/magnitudeU * magnitudeV -- Properties of the DOT product: 1.  commutative: u * v = v * u 2.  squared u * u does not equal u^2 u * u = magnitude of u^2 3.  K(u * v)= Ku * v doesn't distribute 4.  u * (v+w) = u * v + u * w _____________________________________________________________________ Ex.1: u(3, -6) v(4,2) w(-12,-6) -Find the u*v and v*w. Show that u and v are orthogonal and v and w are parallel u*v = 3(4) + -6(2) = 0 which means they are perpendicular because it equals 0 v*w = -12(2) + 2(-6) = -60 w/v = -12/4 = -3 -6/2 = -3 : They are multiples of each other therefore they are parallel

Dot Product

Dot Product looks like this:
u * v = <x1, y1> * <x2, y2> = x1x2 + y1y2
If it equals 0, then the vectors are perpendicular.
If the vectors are multiples of each other then they're parallel.
Properties of Dot Products:
1. Commutative: u * v = v * u
2. Squared: u* u =! u^2; u * u = |u^2|
3. k(u * v) = ku * v
4. u * u = |u|^2
To find angle b/t two vectors: cos(theta) = u * v / |u| |v|
Examples:
_____________________________________________

1.) Given u = (3, -6), v = (4, 2), w = (-12, -6), find u · v and v · w
and show that u and v are perpendicular and v and w are parallel.

(3)(4) + (-6)(2)
12 + (-12) = 0
-12/4 = -3


(4)(-12) + (2)(-6)
-48 + (-12) = -60
-6/2 = -3

Cool Stuff About Math

A trig chart is a derived out list of trig functions. On there is six trig functions: sine (sin) cosine (cos) secant (sec) cosecant (csc) tangent (tan) and cotangent (cot). You can use the trig chart to evaluate angles in degrees or radians.

Trig Chart
(bold applies for all six functions)
sin 0 = 0 cos=1
sin 30 (pie/6) = 1/2 =sqaure root3/3
sin 45 (pie/4) =square root2/2 =square root2/2
sin 60 (pie/3) =square root3/3 =1/2
sin 90 (pie/2) =1 =0

scs=undefined sec=1
=2 =2 square root 3/3
=square root 2 =square root 2
=2 square root 3/3 =2
=1 =undefined

tan=0 
cot=undefined=
3 square root3/3 =square root 3
=1 =1
=sqaure root 3 =3 sqare root3/3
=undefined =0

ex:using trig chart evaluate
A. cos 30 B. tan pie/4
square root3/3 1

vectors

Find u*v and v*w.  Show that u and v are orthogonal and v and w are parallel.
u=(3,-6) v=(4,2) w=(-12,-6)
u*v=3(4)+-6(2)=0
v*w=4(-12)+2(-6)=-60
w/v=-12/4=-3
        -6/2=-3
We are given 3 different points:u,v, and w.  First it tells you to find u*v and v*w.  These are called dot products.  To solve a dot product, you have to multiply your x1 and x2 together and your y1 and y2 together.  You get those two products and add them together to get your answer.  If the dot product equals 0, then the vectors are orthogonal.  Orthogonal means perpendicular.  The dot product of u and v equals zero, so that product is orthogonal.  Now it says to show that v and w are parallel.  If the vectors are multiples of each other they are parallel.  When you divide -12 by 4, you get -3, and when you divide -6 by 2, you get -3 also.  -3 are multiples of each other, so v and w are parallel.

Dot Product

Dot Product: u*v = <x1,y1> * <x2,y2> = x1 x2 + y1 y2
-If the dot produect equals 0 thent he vecotrs are orthogonal (perpendicular)
-If the vectors are multiples fo each other they are parallel
-To find the angle b/w two vectors cos theta = u*v over the magnitude of u and v


Example 1: u(3, -6) v(4,2) w(-12,-6)
-Find the u*v and v*w. Show that u and v are orthogonal and v and w are parallel

u*v = 3(4) + -6(2) = 0 which means they are perpendicular because it equals 0
v*w = -12(2) + 2(-6) = -60

w/v = -12/4 = -3
-6/2 = -3 : They are multiples of each other therefore they are parallel

Example 2: To the nearest degree find the angle between the vectors (1,2) and (-3,1)

u*v = 1(-3) + 2(1) = -1

magnitude of u = square root of 5
magnitude of v = the square root of 10

-Now following the formula in point 3 we need to do cos theta
1/ square root of 5 times square root of 10
theata = inverse of cos(-1/ square root of 50) = 81.867
-You find it in the 1st and 3rd quadrents: 98.130 and 261.87

--Daniellee

-The arithmetic mean of number's a and b = a+b/2 -The geometric mean of number's a and b = square root of ab Note: A sequence is a list of numbers. An arithmetic sequence is a sequence that is generated by adding the same number each time. Arithmetic Sequence formula : tn=t1 + (n-1)d t1= first term n= term number d= difference  tn= ___ term A geometric Sequence is a sequence that is generated by multiplying the same number each time. (to divide, we use fractions) formula: tn=t1(r)^(n-1) t1= first term r= what is being multiplied n= term number tn= ___ term Example: 1.) Find the first four terms of the sequence and state whether the sequence is arithmetic, geometric, or neither. tn= 4n+12 n=1 4(1)+12=16 n=2 4(2)+12=20 n=3 4(3)+12=24 n=4 4(4)+12=28 It is arithmetic because you are adding 4 each time! Example: 2.) Find the first four terms of the sequence and state whether the sequence is arithmetic, geometric, or neither. tn=-2n+8 n=1 -2(1)+8=6 n=2 -2(2)+8=4 n=3 -2(3)+8=2 n=4 -2(4)+8=0 It is arithmetic because you are adding -2 each time!

DOT Product!

Hello,I hope everyone had a great weekend. Im currently stressing over the trig exam this week; I'm sure I am not the only one. 
Anyway,

DOT Product:
u*v=<x1, y1> * <x2, y2> = x1x2+y1y2

-- If the dot product equals zero, then the vectors are orthogonal, which means perpendicular.
-- If the vectors are multiples of each other, then they are parallel.
-- To find the angle between two vectors, use the formula:
cos(theta)= u*v/magnitudeU * magnitudeV

-- Properties of the DOT product:
1.  commutative:
u * v = v * u
2.  squared
u * u does not equal u^2
u * u = magnitude of u^2
3.  K(u * v)= Ku * v
doesn't distribute
4.  u * (v+w) = u * v + u * w

Example 1:
Find (2,3) * (4,-5)
2(4)+3(-5)
=-7

Example 2:
Find (3,-5) * (7,4)
3(7)+-5(4)
=1

Example 3:
Find (-3,0) * (5,7)
-3(5)+0(7)
=-15

Example 4:
Find (3/5,4/5) * (1/2,-3/2)
3/5(1/2)+4/5(-3/2)
=-9/10

Example 5:
Find the value of a if the vectors (6,-8) and (4,a) are parallel.
a/4=3/6
6a=12
divide both sides by 6
a=2

Example 6:
If u=(-2,3), find u*u
-2(-2)+3(3)
=13

Example 7:
If u=(5,-3) and v=(3,7), verify that u * v = v * u
5(3)+-3(7)=-6
3(5)+7(-3)=-6
-6=-6

Example 8:
If u=(5,-3) and v=(3,7), verify that 2(u * v) = (2u) * v
2(-6)= -12
10(3)+ -6(7)=-12
-12=-12

The dot product

If the dot product equals zero, the vectors are perpendicular.  If the vectors are multiples of each other, they are parallel.  Formula for dot product:  u · v = · = x1x2 + y1y2 You  use this formula to find the angle between two vectors: cosƟ = u · v/|u||v|   Properties of the dot product 1. Commutative  u · v = v · u u · u ≠ u^2 2. Squared  u · u = |u|^2 3. k(u · v) = ku · v  doesn't distribute 4. u(v + w) = u  · v + u · w Example 1- V=2i+4j   W=i+5j v · w  =  (2)(1) + (4)(5)  =  22 Example 2- Find the angle between  v=  2i + 3j + k and w  =  4i + j + 2k. ||v||=√4+9+1=√14 ||w||=√16+1+4=√21 v . w  =  8 + 3 + 2 = 13 Ɵ=cosƟ^-1((13)/(√14)(√21))

double and half angles O.o oooooo

sin2α= 2sinαcosα

cos2α= cos²α -sin²α

cos2α= 1 -2sin²α

cos2α= 2cos²α -1

tan2α= 2tanα/1 -tan²α

sin(α/2)= ±√((1 - cosα)/2)

cos(α/2)= ± √((1 + cosα)/2)

tan(α/2)= ±√(1 - cosα)/(1+cosα)

tan(α/2)= sinα/(1 + cosα)

tan(α/2)= (1 - cosα)/sinα

the plus or minus things are determined by where the original angle is.

Example 1

Find the exact value of sin120° (Hint... use your trig chart.).

sin120°= sin2(60°)

sin120°= 2sin60°cos60°

sin120°= 2(½)*√(3)/2

sin120° = √(3)/2

Example 2 

Find the exact value of cos15° (Hint... use the trig chart.).

cos15°= cos(30°)/2

cos15°= ± √(1 + cos30°)/2

cos15°= ± √(1+½)/2

cos15°= ± √(3/2)/2

cos15°= ±√3/4

cos15°= ±(√3)/2

cos15°= (√3)/2

Example 3

Simplify as much as possible:

sinΘtanΘ + cos2ΘsecΘ

convert everything to sine and cosine

sinΘ(sinΘ/cosΘ)+cos2Θ(1/cosΘ)

change cos2Θ to cos²Θ -sin²Θ

(sin²Θ/cosΘ) + ((cos²Θ -sin²Θ)/cosΘ))

algebra

(sin²Θ/cosΘ) + cosΘ -(sin²Θ/cosΘ)

cancel

cosΘ 

The Spontaneous Dot Product

 Today's lesson is on the Dot Product.

First here are some key notes you need to know about the dot product

• If the dot product equals zero, then the vectors are orthogonal, which means perpendicular. If the vectors are multiples of each other, then they are parallel. 
• When finding the angle between two vectors, use the formula: cos(theta)= u*v/magnitudeU * magnitudeV

Now how do we find the dot product:

• First you have two points.
• To find the product of something means to multiply.
• u*v= (x1)(x2)+(y1)(y2) is the dot product formula.

Properties of the Dot Product:

• 1.  commutative: u * v = v * u
• 2.  squared: u * u = magnitude of u^2 
• 3.  K(u * v)= Ku * v does not distribute. "I had problems with this on my test LOL."
• 4.  u * (v+w) = u * v + u * w

Examples:

• Find (3,2) * (5,-4): 3(5)+2(-4) =7
• Find (3/4,2/4) * (1/2,-5/2): 3/4(1/2)+2/4/(-5/2) =0 =orthogonal
• Find the value of (a) if the vectors (12,a) and (6,3) are parallel. a/4=6/8 6a=12 divide both sides by 8 a=3

That's the basics of the Dot product.

Sum and Difference Formulas

sin(x+y) = sin(x)cos(y) + cos(x)sin(y)
sin(x-y) = sin(x)cos(y) - cos(x)sin(y)


cos(x+y) = cos(x)cos(y) - sin(x)sin(y)
cos(x-y) = cos(x)cos(y) + sin(x)sin(y)


tan(x+y) = [tan(x) + tan(y)] / [1-tan(x)tan(y)]
tan(x-y) = [tan(x) - tan(y)] / [1+tan(x)tan(y)]


These formulas are used to find the exact value of something that isn't on the trig chart, like cos(15°).You will use only angles on the trig chart to plug into the formulas. (30°,45°,60°,90°, and any angle that has a reference angle on the trig chart)


Ex. 1cos(15°)

45-30 = 15

cos(45-30) = cos(45)cos(30) + sin(45)sin(30)

cos(15°) = (√(2)/2)*(√(3)/2) + (√(2)/2)*(1/2)

cos(15°) = (√(6)/4) + (√(2)/4)

cos(15°) = [√(6)+√(2)] /4


Ex. 2tan(75°)

45 + 30 = 75

tan(45+30) = [tan(45) + tan(30)] / [1 - tan(45)tan(30)]

tan(75) = [1 + (√(3)/3)] / [1 - (1)*(√(3)/3)]

tan(75) = [(3+√(3))/3)] / [(3 - √(3))/3)]

tan(75) = [9+√(3)] / [9-√(3)]

tan(75) = [81 + 18√(3) + 3] / [81-3]

tan(75) = [84 + 18√(3)] / [78]

tan(75) = [14 + 3√(3)] / [13]


Ex. 3sin(105°)

45+60 = 105

sin(45+60) = sin(45)cos(60) + cos(45)sin(60)

sin(105) = (√(2)/2)*(1/2) + (√(2)/2)*(√(3)/2)

sin(105) = (√(2)/4) + (√(6)/4)

sin(105) = [√(2)+√(6)] /4

Dot Product

u · v = <x1, y1> · <x2, y2> = x1x2 + y1y2

If the dot product = 0 then the vectors are perpendicular

If the vectors are multiples of each other they are parallel

To find the angle between 2 vectors cosƟ = u · v/|u||v|

Properties of the dot product

1. Commutative 2. Squared 3. k(u · v) = ku · v

u · v = v · u u · u ≠ u^2 doesn't distribute

u · u = |u|^2

4. u(v + w) = u · v + u · w

Ex 1. u = (3, -6) v = (4, 2) w = (-12, -6)

Find u · v and v · w. Show that u and v are perpendicular

and v and w are parallel

(3)(4)+ (-6)(2) (4)(-12) + (2)(-6)

12 + (-12) = 0 -48 + (-12) = -60

-12/4 = -3 -6/2 = -3

Ex 2. To the nearest degree find the angle between the

vectors (1, 2) and (-3, 1)

cosƟ = u · v/|u||v u · v = (1)(-3) + (2)(1) = -1

|u| = sqrt1^2 + 2^2 |v| = sqrt-3^2 + 1^2

= sqrt5 = sqrt10

cosƟ = -1/sqrt5 · sqrt10

Ɵ = cos^-1(-1/sqrt50) = 81.867

81.867 + 180 = 261.8 degrees

180 – 81.867 = 98.130 degrees

Ɵ = 98 degrees, 262 degrees

Reference Angles

Reference Angles are like reduced fractions. They must be between 0degrees and 90degrees.

Steps:
1. Find the original quadrant and sketch.
2. Determine if the angle is positive or negative using unit circle methods.
3. Subtract 360degrees or 180degrees until theta is between 0degrees and 90degrees.

Example 1:
Express each of the following in terms of a reference angle.
a. sin128degrees
subtract 180
=52
sin 128 is positive in the second quadrant
=sin52degrees
b. cos128degrees
subtract 180
=52
cos128 is negative in the second quadrant
=-cos52degrees

Example 2:
Express each of the following in terms of a reference angle.
a. sin310degrees
subtract 360
=50
sin310 is negative in the fourth quadrant
=-sin50degrees
b. sin1000degrees
subtract 360
=640degrees
subtract 360
=sin280
subtract 180
=80
sin280degrees is negative in the fourth quadrant
=-sin80degrees

Example 3:
Express each of the following in terms of a reference angle.
a. cos224.5degrees
subtract 180
=44.5
cos 224.5 is negative in the third quadrant

Inverses

Find the inverse of f(x)=2x-3
x=2y-3
y=x+3/2

To have an inverse that is a function it must pass the horizontal line test. The horizontal line test is when you draw a line across your paper and if your equation that you are given hits the line more than one time, your equation can not have an inverse. If the line only crosses your equation one time, then it can have an inverse and you can solve the equation. To find the inverse you switch the x and y in your equation given and then you solve for y. So for this equation I am given, I used the horizontal line test and since this equation is linear, it will pass the horizontal line test because it only crosses once. To solve this equation, I switched the x and y and then I added 3 to both sides and got 2y=x+3. Then I divided everything by 2 and I got y=x+3/2. So the inverse of this function is y=x+3/2.

Cool Formulas You May Want to Know

Chapter 7
-Convert degrees to radians: degrees = pi over 180
-Convert radians to degrees: rads x 180 over pi = degrees
-k= 1/2r^2(theta)
-k= 1/2rs
-s= r(theta)
-Unit Circle= extremely important!

Chapter 8
-m= tan(alpha)
-csc(theta)=1/sin(theta)
-sec(theta)=1/cos(theta)
-cot(theta)=1/tan(theta)
-sin^2(theta) + cos^2(theta) = 1
-1 + tan^2(theta) = sec^2(theta)
-1 + cot^2(theta) = csc^2(theta)

Chapter 9
-SOHCAHTOA
-sin(theta)= opp/hyp
-cos(theta)= adj/hyp
-tan(theta)= opp/adj
-sin A/a = sinB/b = sinC/c
-Right triangle: 1/2bh
-Non right triangle: 1/2(adj leg)(adj leg)sin(angle below)
-Law of Cosines: leg^2= adj leg^2 + other adj leg^2 - 2(adj leg)(other leg)cos(angle below)

Chapter 10
-cos(alpha plus or minus beta) = cos(alpha)cos(beta) minus or plus sin(alpha)sin(beta)
-sin(alpah plus or minus beta)= sin(alpha)cos(beta) plus or minus cos(alpha)sin(beta)
-tan(alpha+beta) = tan(alpha + beta)/1-tan(alpha)tan(beta)
-tan(alpha-beta) = tan(alpha-beta)/1+tan(alpha)tan(beta)
-sin2(alpha)= 2sin(alpha)cos(alpha)
-cos2(alpha)= cos^2(alpha)-sin^2(alpha)
-cos2(alpha)= 1-2sin^2(alpha)
-cos2(alpha)= 2cos^2(alpha)-1
-tan2(alpha)= 2tan(alpha)/1-tan^2(alpha)
-sin(alpha)/2= plus or minus square root of 1-cos(alpha)/2
-cos(alpha)/2= plus or minus square root of 1+cos(alpha)/2
-tan(alpha)/2= plus or minus square root of 1-cos(alpha)/1+cos(alpha)
-tan(alpha)/2= sin(alpha)/1+cos(alpha)
-tan(alpha)/2= 1-cos(alpha)/sin(alpha)

Some Cool Stuff To Know

tan(α + β) = tanα + tanβ/1 – tanαtanβ – Sum

tan(α - β) = tanα - tanβ/1 + tanαtanβ – Difference

Example 1.

 Suppose tanα = 1/3 and tanβ = ½

a. Find tan(α + β)

tanα + tanβ/1 – tanαtanβ = 1/3 + ½ /1 – (1/3)(1/2)

2/6 + 3/6 / 1 – 1/6 = 5/6 / 5/6 = 1

b. Find tan(α – β)

tanα - tanβ/1 + tanαtanβ = 1/3 – ½ /1 + (1/3)(1/2)

2/6 – 3/6 / 1 + 1/6 = -1/6 / 7/6 = -6/42 = -1/7

Example 2.

 tanα = 2/3 and tanβ = ½

a. tanα + tanβ/1 – tanαtanβ = 2/3 + ½ /1 – (2/3)(1/2)

4/6 + 3/6 / 1 – 2/6 = 7/6 / 2/3 = 21/12 = 7/4

b. tanα - tanβ/1 + tanαtanβ = 2/3 – ½ /1 + (2/3)(1/2)

4/6 – 3/6 / 1 + 2/6 = 1/6 / 4/3 = 3/24 = 1/8

Domain and Range

Domain-the interval of x values where the graph exists.
Rage-the interval of y values where the graph exists.

For zeros x-intercept, root -set = o. Solve for x.
To be a function the graph must pass the vertical line test. For this you draw lines up and down to see if part of the graph crosses the line twice. If if crosses any line more than once it is not a function.
*If given points, the domain is the list of all x values, while range is a list of all y values in a set of points indicated.

1.Polynomials
Domain: 00 or -00 always
Range: odd (-oo or 00). X^2 always (-b/2a, 00) or (-oo, -b/2a)

2.Fractions
Domain: -factor top and bottom
-cancel if possible (mark #)
-set bottom = 0, then solve the x value
-write in interval notation stopping at #s found in { }
Range: is none

Example1.

Find the domain and range of 3+2x^5
Domain = -oo, oo
Range= -oo, 00

Example2.

Is the following a function? (2,-3) (4,-2) (2,2) , (3, -2)
No because when it is drawn out in a graph it does not pass the vertical line test.

Example3.

(x-2) (x-3)/(x-6) (x-3) Find domain and range
-cancel the pay attenction only to top numbers
Domain= (-oo, 3) u (3,6) u (6, 00)
Range= none

Vector Explanations with Rico Gathers

Today we are going to talk about Vectors.

First what's a vector?
 -a vector is a slope. It contains an x-component along with an Y-component.

There are many ways to find vectors. Here's two of the more simple ones. Although all are pretty easy.

1) Vector Addition : V+U= <a,b> + <c,d> = <a+c,b+d>

        -Ex. (1,9) + (4,5)
        
             (1+4,9+5) = <5,14>

2) Vector Subtraction : V-U= <a,b> - <c,d> = <a-c,b-d>


            -Ex. (1,9) - (4,5)
        
             (1-4,9-5) = <-3,4>

That's our lesson for today with Rico Gathers ;-).

Formulas and Vector Equations

I'm bored, so I thought I'd get started a little early :)

Distance: Square root of (x2-x1)^2 + (y2-y1)^2 + (z2+z1)^2

Midpoint: (x1+x2/2  y1+y2/2  z2+z1/2

Sphere: (x-xo)^2 + (y-yo)^2 + (z-zo)^2 = r^2

Vector Equation: (x,y,z) = (xo,yo,zo) + t(a,b,c)

Parametric: x= xo+at  y=yo+bt  z=zo+ct

Magnitude of U = square root of a^2+b^2+c^2

EXAMPLE1: With the give points, find the vector equation and parametric (2,3) (0,1)

(2,3) + (0,1)

= (2,4)

(x,y)= (2,3) + t(2,4) 

x= 2 + 2t

y= 3 + 4t

step 1: Add the points to find your vector

step 2: Plug your first given point and your new vector straight into the formula

step 3: x= the x value of your first given point and the a value of the new vector with t behind it

step 4: y= the y value of your first given point and the b value of the new vector with t behind it 

EXAMPLE2: Find the magnitude of u (3,4)

square root of 3^2 + 4^2

= square root of 9 + 16

= square root of 25

=5

step 1: plug given points into your magnitude formula (square root of x^2 + y^2)

step 2: 3x3 + 4x4

step 3: after you multiply, then add, you are going to get 25

step 4: square root 25 to get your final answer of 5

EXAMPLE 3: Find a vector equation of the line through A(3,4) & B(5,5)

(3,4) + (5,5) = (8,9)

(x,y)= (3,4) + t(8,9)

Step1: Add points to find vector.

Step2: Plug your first given point and your new vector directly into the formula 

This is my blog for this week. Vectors A vector is a slope. vector addition: v+u=+= vector subtraction: v-u= -= vector multiplication: kv=k= To find a vector from two points: p2-p1 Vector equation: (x,y)=(x0, y0)+t(a,b) Parametic equations: x=x0+at y=y0+bt absolute value of v= square root of x^2+y^2= magnitude of a vector Example 1: Give the component form of Ab and find the magnitude of AB when A(1,-2), B(3,-2) (3,-2)-(1,-2)= (3-1,-2+2) = <2,0> vector square root of -4^2+-3^2= square root of 16+9= square root of 25 = 5 magnitude Example 2: Let u= (3,1) and v= (-8,4). Find u+v. (3+-8, 1+4) =(-5,5) Example 3: Let u= (3,1) v= (-8,4) and w= (-6,-2). Calculate each expression. a. u+v (3+-8, 1+4) =<-5,5) b. u-v <3+8, 1-4> =<11,-3> c. 3u+w <9,3> + <-6,-2> =<3,1> Have a lovely day.

vectos

Vector addition - v + u = + =

Vector subtraction - v – u = - =

Vector multiplication – kv = k =

To find a vector from 2 points do P2 – P1

Vector equation - (x, y) = (x˳, y˳) + t(a, b)

Parametric equations:

x = x˳ + at

y = y˳ + bt 

magnitude of a vector

-|u| = sqrt x^2 + y^2     

Example 1: Finding the magnitude/ lenght

http://www.youtube.com/watch?v=6GoMXuE1FOw

Example 2: Addition and subtraction of vectors

http://www.youtube.com/watch?v=3yxYAy7cCBQ&feature=related