The Cross Product and Oriented Area · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

The product that points sideways

Mathematics · Geometry · ages 19-20
Name ______________________   Date ____________

  1. Ava says u cross v is perpendicular to both u and v. Is Ava right?

    Circle one:   True   False

  2. Ana says u cross u is the zero vector for every vector u. Is Ana right?

    Circle one:   True   False

  3. Compute (1, 1, 0) cross (0, 1, 1).

    • (1, -1, 1)
    • (-1, 1, -1)
    • (1, 1, 1)
    • (0, 1, 0)

  4. What is j cross k?

    • 0 (zero vector)
    • -i
    • i

  5. Kai computes (1, 2, 3) cross (4, 5, 6) as (-3, 6, -3), then dots that result with (1, 2, 3) and gets 0. Is his result perpendicular to (1, 2, 3)?

    Circle one:   True   False

  6. A triangle has vertices (1, 1, 1), (2, 1, 1) and (1, 4, 5). What is its area?

    Answer: ______________

  7. A parallelogram has edges (1, 2, 0) and (3, 1, 0). What is its area?

    Answer: ______________

  8. What is j cross i?

    • -k
    • k
    • 0 (zero vector)

  9. Compute (1, 2, 0) cross (0, 1, 3).

    • (6, 3, 1)
    • (6, -3, 1)
    • (-6, 3, -1)

  10. Leo says v cross u equals minus (u cross v) for any 3D vectors u and v. Is Leo right?

    Circle one:   True   False

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Answer key

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

The product that points sideways W1-mt_QEBM_Ms-SK-s1

  1. True · Dotting the cross with either input gives zero, which means a right angle both ways.
  2. True · Swapping u with itself changes nothing, yet the swap rule says the result negates. Only the zero vector equals its own negative.
  3. (1, -1, 1) · The formula gives (1 times 1 - 0 times 1, 0 times 0 - 1 times 1, 1 times 1 - 1 times 0) = (1, -1, 1).
  4. i · j to k runs in order around the cycle i, j, k, i, so j cross k is the next vector, i.
  5. True · The formula gives (12 - 15, 12 - 6, 5 - 8) = (-3, 6, -3). Its dot with (1, 2, 3) is -3 + 12 - 9 = 0.
  6. 2.5 · The edges from (1, 1, 1) are (1, 0, 0) and (0, 3, 4). Their cross is (0, -4, 3) of length 5, and the triangle takes half: 2.5.
  7. 5 · The cross product is (0, 0, -5), whose length is 5.
  8. -k · Swapping i cross j = k reverses the sign, giving minus k.
  9. (6, -3, 1) · The formula gives (2 times 3 - 0 times 1, 0 times 0 - 1 times 3, 1 times 1 - 2 times 0) = (6, -3, 1).
  10. True · The cross product is antisymmetric: swapping inputs flips the sign.
Worksheet · LightMySky