Questions: Dot and Cross Products: Geometry and Computation

Question 1

You need to find a vector that is perpendicular to both u = (1, 0, 0) and v = (0, 1, 0). Which operation produces this directly?

• A. Compute u · v and normalize the result
• B. Compute |u||v|cos θ to find the angle, then rotate
• C. Compute u × v
• D. Add u + v and find its magnitude

Answer: (C)

The cross product u × v produces a vector perpendicular to both u and v. Here, u × v = (0,0,1) = k, which is indeed perpendicular to both. The dot product (option A) produces a scalar, not a vector, so it cannot give a perpendicular direction.

Question 2

You compute u · v and get 0. Your friend says: 'The dot product is zero, so one of the vectors must be the zero vector.' Is your friend correct?

• A. Yes — a dot product of zero requires at least one zero vector
• B. No — u · v = 0 means u and v are perpendicular (orthogonal), not that either is zero
• C. No — u · v = 0 means u and v are parallel and point in opposite directions
• D. Yes — the dot product equals |u||v|, so if it is 0 then at least one magnitude is 0

Answer: (B)

u · v = |u||v|cos θ = 0 when cos θ = 0, i.e., θ = 90°. Two non-zero vectors can have zero dot product if they are perpendicular. Your friend is confusing the product of scalars with the dot product: 0 = |u||v|cos θ is satisfied when either vector is zero OR when the vectors are orthogonal. The key insight is that orthogonality is the primary geometric meaning of a zero dot product.

Question 3

The magnitude of the cross product u × v equals the area of the parallelogram whose sides are u and v.

• T. True
• F. False

Answer: True

|u × v| = |u||v|sin θ, where θ is the angle between u and v. The area of a parallelogram with sides u and v is also base × height = |u| × (|v|sin θ) = |u||v|sin θ. These are equal, so the cross product magnitude directly encodes the parallelogram area.

Question 4

If u × v = 0, then at least one of u or v should be the zero vector.

• T. True
• F. False

Answer: False

u × v = 0 when |u||v|sin θ = 0, which happens either when one vector is zero OR when sin θ = 0, i.e., θ = 0° or 180°. Two parallel (or antiparallel) non-zero vectors have zero cross product because they span a parallelogram of zero area — there is no well-defined perpendicular direction. For example, u = (1,0,0) and v = (2,0,0) give u × v = 0 even though both are non-zero.

Question 5

A student says: 'The dot product tells me the angle between two vectors.' Is this correct? What does the dot product directly measure, and how would you obtain the angle from it?

Model answer: The statement is incomplete. The dot product u · v = |u||v|cos θ encodes the angle, but it also depends on the magnitudes of both vectors. The dot product directly measures 'alignment weighted by magnitude' — how much the vectors point in the same direction, scaled by their lengths. To extract the angle alone, you must divide: cos θ = (u · v) / (|u||v|). If you want only whether two vectors are perpendicular, the dot product is sufficient (zero = perpendicular); if you want the actual angle in degrees, you need to normalize by the magnitudes first.

This distinction matters in applications. The projection of u onto v is (u · v)/|v|, which uses the dot product divided by one magnitude. The angle formula divides by both. Students who skip the normalization step get an answer that conflates direction with scale.