10.3 dot product (“multiplying vectors”) properties of the dot product angle between two vectors...

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10.3 Dot Product (“multiplying vectors”) Properties of the dot product Angle between two vectors using dot product Direction Cosines Projection of a vector onto another vector Work done by constant force

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Page 1: 10.3 Dot Product (“multiplying vectors”) Properties of the dot product Angle between two vectors using dot product Direction Cosines Projection of a vector

10.3 Dot Product (“multiplying vectors”)

Properties of the dot product Angle between two vectors using dot product

Direction CosinesProjection of a vector onto another vector

Work done by constant force

Page 2: 10.3 Dot Product (“multiplying vectors”) Properties of the dot product Angle between two vectors using dot product Direction Cosines Projection of a vector

Page 3: 10.3 Dot Product (“multiplying vectors”) Properties of the dot product Angle between two vectors using dot product Direction Cosines Projection of a vector

Page 4: 10.3 Dot Product (“multiplying vectors”) Properties of the dot product Angle between two vectors using dot product Direction Cosines Projection of a vector

The angle between two nonzero vectors with the same initial point is the smallest angle between them.

Page 5: 10.3 Dot Product (“multiplying vectors”) Properties of the dot product Angle between two vectors using dot product Direction Cosines Projection of a vector

Find the angle between the vectors v = <2, 1, -1> and w = <3, -4, 1>

Find the angle between the vectors v = <-2, 2, 1> and w = <2, 3, 6>

Page 6: 10.3 Dot Product (“multiplying vectors”) Properties of the dot product Angle between two vectors using dot product Direction Cosines Projection of a vector

Find the angle between the two vectors F1 and F2 where F1 = j-k and F2 = 2i-j+2k

Page 7: 10.3 Dot Product (“multiplying vectors”) Properties of the dot product Angle between two vectors using dot product Direction Cosines Projection of a vector

Page 8: 10.3 Dot Product (“multiplying vectors”) Properties of the dot product Angle between two vectors using dot product Direction Cosines Projection of a vector

Find a number k such that u = <2, 3, 4> is orthogonal to v = <k, 3, -7>

Page 9: 10.3 Dot Product (“multiplying vectors”) Properties of the dot product Angle between two vectors using dot product Direction Cosines Projection of a vector

Page 10: 10.3 Dot Product (“multiplying vectors”) Properties of the dot product Angle between two vectors using dot product Direction Cosines Projection of a vector

Page 11: 10.3 Dot Product (“multiplying vectors”) Properties of the dot product Angle between two vectors using dot product Direction Cosines Projection of a vector

Page 12: 10.3 Dot Product (“multiplying vectors”) Properties of the dot product Angle between two vectors using dot product Direction Cosines Projection of a vector

Page 13: 10.3 Dot Product (“multiplying vectors”) Properties of the dot product Angle between two vectors using dot product Direction Cosines Projection of a vector

Page 14: 10.3 Dot Product (“multiplying vectors”) Properties of the dot product Angle between two vectors using dot product Direction Cosines Projection of a vector

Find the direction cosines and angles of the vector v = (4, -2, -4)

Page 15: 10.3 Dot Product (“multiplying vectors”) Properties of the dot product Angle between two vectors using dot product Direction Cosines Projection of a vector

Page 16: 10.3 Dot Product (“multiplying vectors”) Properties of the dot product Angle between two vectors using dot product Direction Cosines Projection of a vector

Normal Vector (orthogonal to PQ)

Page 17: 10.3 Dot Product (“multiplying vectors”) Properties of the dot product Angle between two vectors using dot product Direction Cosines Projection of a vector

Page 18: 10.3 Dot Product (“multiplying vectors”) Properties of the dot product Angle between two vectors using dot product Direction Cosines Projection of a vector

Ex 2: Given a = <2, -6, 3> and b = <1, -2, -2>, find the vector projection of b onto a.

Page 19: 10.3 Dot Product (“multiplying vectors”) Properties of the dot product Angle between two vectors using dot product Direction Cosines Projection of a vector

Which of the following does not make sense?

Page 20: 10.3 Dot Product (“multiplying vectors”) Properties of the dot product Angle between two vectors using dot product Direction Cosines Projection of a vector

WORK: YOU HAVE TO CHANGE KINETIC ENERGY OF AN OBJECT TO DO WORK! IN OTHER WORDS: EXERTING FORCE DOES NOT EQUAL WORK!

Physics

Page 21: 10.3 Dot Product (“multiplying vectors”) Properties of the dot product Angle between two vectors using dot product Direction Cosines Projection of a vector

Application #1

A woman exerts a horizontal force of 25lb on a crate as she pushes it up a ramp that is 10 feet long and inclined at an angle of 20 degrees above the horizontal. Find the work done on the box.

Page 22: 10.3 Dot Product (“multiplying vectors”) Properties of the dot product Angle between two vectors using dot product Direction Cosines Projection of a vector

Applications in real life #2: work

W = (magnitude of force) (displacement)= |F||D|cos(theta)

A mass is dragged up an incline of 38 degrees for 2 m by a force of 5.8 N that is directed at an angle of 54 degrees to the horizontal as shown in the diagram. What is the work done?

Page 23: 10.3 Dot Product (“multiplying vectors”) Properties of the dot product Angle between two vectors using dot product Direction Cosines Projection of a vector

SEC. 10.3/ 1, 3, 9, 10, 11, 13, 15, 19, 21, 27, 31, 45, 69, 70

Homework/Classwork

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