1. ConcepTest Section 13.3 Question 1 View the second and hour hands of a clock as vectors and in the plane. Describe over the course of 2 minutes beginning at noon. If you increase the length of the second hand, what happens to the dot products?

2. ConcepTest Section 13.3 Answer 1 Over the course of two minutes, the hour hand moves very little, while the second hand makes two complete revolutions. The dot product starts out at a maximum with both vectors pointing at 12. As moves, the dot product decreases until it is zero when the second hand is near 3, it keeps decreasing and becomes negative until it reaches a minimum when points approximately toward 6. After that, the dot product increases, reaches zero when points approximately at 9 and keeps increasing until it reaches a maximum when points at about 12. During the second minute the dot product repeats the same pattern with slight variation due to the small movement of the hour hand. If the length of is increased, the dot product goes through the same sign changes and has the same zeros but reaches a larger maximum value and a lower (negative) minimum. ANSWER COMMENT: This is a good place to reinforce that the dot product depends on both the magnitude and the angle between the vectors.

3. ConcepTest Section 13.3 Question 2 Which of the following planes are parallel to one another? Which are perpendicular to one another?

4. ConcepTest Section 13.3 Answer 2 (a) and (b) are perpendicular, (a) and (f) are perpendicular, (c) and (e) are parallel. ANSWER COMMENT: Follow-up Question. Does this mean that (b) is perpendicular to (f)? Why not?

5. ConcepTest Section 13.3 Question 3 Draw a picture and describe a situation where (a) The projection of a vector onto a line is the zero vector; (b) The projection of a vector onto a line is the vector itself.

6. ConcepTest Section 13.3 Answer 3 (a) The vector is perpendicular to the line. (b) The vector lies on the line. ANSWER COMMENT: It is important to be able to visualize projections, not just remember the formula.

7. ConcepTest Section 13.3 Question 4 True or false? (a) if and only if (b) The zero vector is orthogonal to any other vector. (c) Any plane has only two distinct normal vectors. (d) If and are large than is large. (e) The dot product of a vector with itself is its magnitude. (f) Any vector normal to a surface has length one. (g) Parallel planes share a same normal vector. (h) If two planes are perpendicular, so are their normal vectors.

8. ConcepTest Section 13.3 Answer 4 (a) False, the vectors could be perpendicular. (b) True (c) False. A plane has infinitely many normal vectors. It has two distinct unit normal vectors. (d) False. Not if the vectors are perpendicular or close to perpendicular. (e) False. It’s the square of its magnitude. (f) False (g) True (h) True ANSWER COMMENT: This problem may bring up students’ misconceptions about vectors.