You want to find the area of a triangle with vertices at the origin O, A = (1, 0, 0), and B = (0, 1, 0). Which expression gives the correct area?
A|A · B| = 0, so the area is 0
B|A × B| / 2 = |(0, 0, 1)| / 2 = 1/2
C|A| · |B| = 1, so the area is 1
D|A · B| / 2 = 0, so the area is 0
The area of the parallelogram spanned by two vectors equals |a × b|, and the triangle is half that parallelogram. A × B = (0·0 − 0·1, 0·0 − 1·0, 1·1 − 0·0) = (0, 0, 1), with magnitude 1, giving triangle area 1/2. Options A and D use the dot product, which measures projection (cosine), not area. Option C uses only the magnitudes, ignoring the angle — it would give the area only if the vectors were perpendicular.
Question 2 Multiple Choice
If a × b = v, what is b × a?
Av (same direction and magnitude as a × b)
B−v (opposite direction, same magnitude)
C0 (reversing order cancels the cross product)
DIt depends on the angle between a and b
The cross product is anti-commutative: b × a = −(a × b) = −v. Swapping the order reverses the orientation of the parallelogram — the right-hand rule now points in the opposite direction. This is a geometric fact about orientation, not a computational quirk. Option A would be true for the dot product, where order doesn't matter because it yields a scalar.
Question 3 True / False
The cross product of two parallel vectors is a nonzero vector pointing perpendicular to both.
TTrue
FFalse
Answer: False
When two vectors are parallel, θ = 0 or π, so sin(θ) = 0, and |a × b| = |a||b|sin(θ) = 0. The cross product is the zero vector. Geometrically, parallel vectors span a degenerate parallelogram (a line segment) with zero area, and there is no unique perpendicular direction to a plane that isn't defined — because no plane is uniquely defined by two parallel vectors.
Question 4 True / False
The magnitude of the cross product a × b equals the area of the parallelogram spanned by a and b, which is why it appears in surface area calculations.
TTrue
FFalse
Answer: True
|a × b| = |a||b|sin(θ) exactly equals base × height of the parallelogram. This is the key to surface area integrals: when a surface is parametrized by r(u, v), the local area element is dS = |rᵤ × rᵥ| du dv, because rᵤ × rᵥ packages both the outward normal direction and the local area scaling factor into a single vector.
Question 5 Short Answer
Why is the cross product anti-commutative (a × b = −(b × a)), and what geometric fact does this algebraic property reflect?
Think about your answer, then reveal below.
Model answer: Swapping the order of vectors reverses the orientation of the parallelogram they define. The right-hand rule determines direction: curl fingers from a toward b for a × b; curl fingers from b toward a for b × a, which points the opposite way. This orientation-dependence reflects the fact that a directed surface changes its 'outward' direction when you reverse the order of its spanning vectors — the same way reversing the traversal direction of a curve switches its orientation.
Anti-commutativity is not an artifact of the determinant formula — it is a geometric statement about orientation in 3D space. The sign records which of the two perpendicular directions is 'selected,' and that selection depends on the order of the inputs.