Solutions to Quiz 2

1. Find an equation of the form n_{1}x + n_{2}y + n_{3}z = d for the plane containing the points P = (1, 2, 3), Q = (2, 4, 8) and R = (2, 3, 5). [5]

Let u⃗ = P⃗ Q =< 1, 2, 5 > and v⃗ = P⃗ R =< 1, 1, 2 >. These are vectors in the plane, so n⃗ = u⃗×v⃗ = (2∗2−5∗1)i⃗+(5∗1−1∗2)j⃗+(1∗1−2∗1)k⃗ =< −1, 3, −1 >. Then the equation of the plane is −x+3y−z = d. Plugging in point P for (x, y, z) yields that d = 2. So an equation of the plane is −x + 3y − z = 2.

2. For which vectors u⃗, v⃗ and w⃗ are the following identities true? Provide brief explanations why you think your answer is true. Your answer should be in one of the forms (say the identity involves only u⃗ and v⃗): [2 each]

For all u⃗, v⃗. For no u⃗, v⃗. Only for u⃗, v⃗ with certain properties. (Name these properties!)

a) (u⃗ · v⃗) · w⃗ = u⃗ · (v⃗ · w⃗).

For no u⃗, v⃗, w⃗. u⃗ · v⃗ is a scalar, so (u⃗ · v⃗) · w⃗ makes no sense, since you can’t dot a scalar with a vector. If you misunderstood the statement as (u⃗ · v⃗)w⃗ = (v⃗ · w⃗)u⃗, then the answer should have been: Only if one of u⃗, v⃗, w⃗ is 0 or if u⃗ and w⃗ are parallel.

b) (u⃗ + v⃗) × w⃗ = u⃗ × w⃗ + v⃗ × w⃗.

For all u⃗, v⃗, w⃗ since the crossproduct is distributive.

c) −u⃗/∥u⃗∥ is a unit vector.

For all u⃗ except 0. For 0 you are dividing by zero, and the expression is not defined. For all other u⃗ we have ∥ − u⃗/∥u⃗∥∥ = ∥ − u⃗∥/∥u⃗∥ = 1 so that the vector is unit. Direction doesn’t matter for unit vectors (unless specified), we only have to have magnitude 1.

d) u⃗ · v⃗ = ∥u⃗∥∥v⃗∥.

u⃗·v⃗ = ∥u⃗∥∥v⃗∥ cos θ. So the equation is only true if either u⃗ or v⃗ is 0, or if cos θ = 1, which means that the vectors are in the same direction (parallel is not enough!).

e) u⃗ × v⃗ = ∥u⃗∥∥v⃗∥ sin θ, where θ is the angle between u⃗ and v⃗.

Not true for any u⃗, v⃗. The left side is a vector and the right side a scalar. It would have been correct for all u⃗, v⃗ if the left side would have been ∥u⃗ × v⃗∥.

3. Two adjacent sides of a regular hexagon are given as the vectors u⃗ and v⃗ below. Label the remaining sides of the hexagon in terms of u⃗ and v⃗. [5]

The vectors are (counterclockwise, starting at u⃗): u⃗, v⃗, v⃗ − u⃗, −u⃗, −v⃗, u⃗ − v⃗.