vectors

Vectors are slopes

to add vectors: v + u =< x, y > + < x1, y1 > = < x+x1, y+y1 >

Scalar Multiplication kv= k < x, y > = < kx, ky >

To find a vector from 2 points: P2-P1

The vector equation is: (x,y) = (xo,yo) + (a,b)

Parametric Equations: x=xo+at ; y=yo+bt

|v| = √ x² + y² = the magnitude of the vector

Component form < rcosΘ, rsinΘ >

a force can be expressed as a vector

Example 1: A force of 2 N acts on angle of 45° with the x-axis find in component form

< 2cos(45°), 2sin(45°) >

=< √2, √2 >

the only step) plug into the formula r is 2 and Θ is 45°

Dot Product

u•v = < X1, Y1> • < X2,Y2 > = X1 X2 + Y1 Y2

-If the dot product equal to 0, then the vectors are orthogonal which is just the really over complicated way of saying perpendicular

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

-To find the angle between 2 vectors use the formula: cosΘ= (u*v) / (|u||v|)

-Properties of the dot products:

Commutative: u*v = v*u

Squared: u*u is not u² u*u = |u²|

k(u*v) = ku*v (does not distribute)

u*(v + w) = u*v + u*w

Example 1: u = (2,-4) v = (6,3)

u•v = 3(4) + (-6)(2) = o

*it is perpendicular

Step1) Plug into formula (see above)

Step2) Multiply and add or plug into calculator if you are lazy