Let u = (1, 0, 0) and v = (0, 1, 0). What is u × v?
A(0, 0, 0)
B(1, 1, 0)
C(0, 0, 1)
D(0, 0, −1)
Applying the component formula: u × v = (u₂v₃ − u₃v₂, u₃v₁ − u₁v₃, u₁v₂ − u₂v₁) = (0·0 − 0·1, 0·0 − 1·0, 1·1 − 0·0) = (0, 0, 1). Geometrically, the cross product of the first and second standard basis vectors is the third — consistent with the right-hand rule.
Question 2 True / False
The cross product is commutative: u × v = v × u for most vectors u and v in R^3.
TTrue
FFalse
Answer: False
The cross product is anti-commutative, meaning v × u = −(u × v). Reversing the order reverses the direction of the resulting vector. This follows from the right-hand rule: curling from v to u points in the opposite direction from curling u to v. This is a fundamental difference from the dot product and from scalar multiplication, both of which are commutative.
Question 3 Short Answer
What geometric quantity does the magnitude ||u × v|| represent?
Think about your answer, then reveal below.
Model answer: The area of the parallelogram with u and v as adjacent sides.
||u × v|| = ||u|| ||v|| sin(θ), which is base times height for the parallelogram formed by u and v. When the vectors are parallel (θ = 0), sin(θ) = 0 and the 'parallelogram' is flat with zero area. When they are perpendicular (θ = 90°), the magnitude reaches its maximum of ||u|| ||v||, matching the area of the rectangle they form.