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Vectors and Angles Lesson 10.3b
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Angle Between Two Vectors
Given vectors v and w form angle We can show that
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Angle Between Two Vectors
Try these two vectors What is the angle between? 1.47 radians or degrees
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Orthogonal Vectors When the dot product equals zero …
What happens to the angle? Orthogonal
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Orthogonal Vectors Are these two vectors orthogonal?
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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)
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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
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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.)
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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.
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Work as a Dot Product Work = Force • Distance
But … what if force not in same direction of movement? F PQ
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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
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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?
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Assignment Lesson 10.3b Page 694
Exercises 11, 13, 31, 33, 35, 37, 39, 41, 45, 49, 51, 53, 55
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