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