The product that points sideways

Mathematics · Geometry · ages 19-20

Name ______________________ Date ____________

Ava says u cross v is perpendicular to both u and v. Is Ava right?
Circle one: True False
Ava says u cross v is perpendicular to both u and v. Is Ava right?
Circle one: True False
Compute (1, 0, 0) cross (0, 1, 0).
- (0, 0, 1)
- (0, 0, -1)
- (1, 1, 0)
- (0, 1, 0)
What is i cross j?
- 0 (zero vector)
- i
- k
Leo says v cross u equals minus (u cross v) for any 3D vectors u and v. Is Leo right?
Circle one: True False
A triangle has vertices (0, 0, 0), (4, 0, 0), (0, 3, 0). What is its area?
Answer: ______________
Vectors u = (2, 0, 0) and v = (0, 3, 0) span a parallelogram. What is its area?
Answer: ______________
What is j cross i?
- -k
- k
- 0 (zero vector)
Compute (1, 0, 0) cross (0, 1, 0).
- (0, 0, -1)
- (0, 0, 1)
- (1, 1, 0)
Leo says v cross u equals minus (u cross v) for any 3D vectors u and v. Is Leo right?
Circle one: True False

Answer key

For grown-ups. Fold this page away before handing over the rest.

True · Dotting the cross with either input gives zero, which means a right angle both ways.
True · Ava is right: dotting the cross with either input gives zero.
(0, 0, 1) · The determinant gives (0 times 0 minus 0 times 1, 0 times 0 minus 1 times 0, 1 times 1 minus 0), which is (0, 0, 1).
k · The basis cycles i cross j = k by the right hand rule.
True · Leo is right: the cross product is antisymmetric, so swapping inputs flips the sign.
6 · Half the cross magnitude: (4, 0, 0) cross (0, 3, 0) = (0, 0, 12), half is 6.
6 · u cross v = (0, 0, 6), whose length is 6.
-k · Swapping i cross j = k reverses the sign, giving minus k.
(0, 0, 1) · The determinant gives (0, 0, 1): ones on x and y push fully into z.
True · The cross product is antisymmetric: swapping inputs flips the sign.