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10.3 Dot Product (“multiplying vectors”)

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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10.3 Dot Product (“multiplying vectors”)

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  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 onto another vector Work done by constant force

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

  3. 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)

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

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

  6. Normal Vector (orthogonal to PQ)

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

  8. Applications in real life: 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?

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

  10. A molecule of methane, CH4, is structured with four hydrogen atoms at the vertices of a regular tetrahedron and the carbon atom at the centroid. The bond angle is the angle formed by H-C-H combination; it is the angle between the lines that join the carbon atom to two of the hydrogen atoms. Show that the bond angle is about 109.5 degrees.[Hint: Take the vertices of the tetrahedron to be the points (1,0,0), (0,1,0), (0,0,1), and (1,1,1) as shown in the figure. Then the centroid is (½, ½, ½).

  11. Which of the following does not make sense?

  12. Homework/Classwork Sec. 10.3/ 1, 3, 9, 10, 11, 13, 15, 19, 21, 27, 31, 45, 69, 70

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