Vectors & Dot Product

• Vector - slope
• Vector addition v + u = <a,b> + <c,d> = <a+c,b+d>
• Scalor Multiplication kv= k<a,b> = <ka,kb>
• To find a vector from 2 points do p2-p1
• Vector equation - (x,y) = (xo,yo) + t(a,b)
• Parametric Equations: x=xo+at ; y=yo+bt
• Negative absolute value of v = square root of x^2+y^2 which = the magnitude of the vector
• Component form < r cos theta, r sin theta >
• FORCE = VECTOR

Example 1: A force of 10 N acts on angle of 130 degrees with the x-axis find in component form

< 10 cos 130 degrees, 10 sin 130 degrees >

= <-6.428, 7.66>

Step1) Draw out your triangle

Step2) Plug givens into your component form formula

Step3) Your 10 is r and 130 is your theta

Dot Product 

U x V = < X1 x Y1 > x < X2 x Y2 > = X1 X2 + Y1 Y2

-If the dot product = o, then the vectors are orthogonal (perpendicular) 

-If the vectors are multiples of each other they are parallel 

-To find the angle between 2 vectors Cos Theta = U x V/ Abs value of U x Abs value of V

-Properties of the dot products: 

1. Communtive .....  U x V = V x U
2. Squared ......  U x U not equal to U^2 ; U x U = Absolute Value of U^2
3. K (U x V) = KU x V (doesn't distribute)
4. U x (V + W) = U x V + U x W

Example 1: U = (3,6) V = (4,2). Find the Dot product of U V

(3,6) (4,2)

12+12 = 24

Step1: Multiply the x's together
Step2: Multiply the y's together
Step3: Add the values together