Vectors and Angles Lesson 10.3b

Angle Between Two Vectors Given vectors v and w form angle  We can show that 

Angle Between Two Vectors Try these two vectors What is the angle between? 1.47 radians or 84.59 degrees

Orthogonal Vectors When the dot product equals zero … What happens to the angle? Orthogonal

Orthogonal Vectors Are these two vectors orthogonal?

Projections Consider v and w, vectors in 2-space v w u = projection of v on w Note that u = t • u (for some scalar, t)

Projections We can say (v – t w) • w = 0 v – t • w (orthogonal to w) v u = projection of v on w We can say (v – t w) • w = 0

This is the scalar projection of v on w. Projections Since (v – t w) • w = 0 This is the scalar projection of v on w. (This is a number.)

The end result is a vector. Projections The vector projection of v in the direction of w is This is a scalar The end result is a vector.

Work as a Dot Product Work = Force • Distance But … what if force not in same direction of movement? F PQ

Work as a Dot Product Example: Wind = F Boat movement = PQ Projection of F on PQ is the force used in direction of movement Work = F • PQ the dot product F PQ

Work as a Dot Product Given F = 2i + 3j + 1k acting on a particle Particle moves from P(1, 0, -1) to Q(3, 1, 2) What is the work accomplished?

Assignment Lesson 10.3b Page 694 Exercises 11, 13, 31, 33, 35, 37, 39, 41, 45, 49, 51, 53, 55