Cross Product Magnitude

∣ a \times b ∣ = ∣ a ∣∣ b ∣ sin θ

The magnitude of a×b is the area of the parallelogram spanned by a and b. The vector a×b is perpendicular to both (right-hand rule). a×a = 0 and a×b = −b×a. Triangle area is (1/2)|a×b|.

$∣ a ∣, ∣ b ∣$: Vector magnitudes
$θ$: Included angle
$a \times b$: Cross-product vector

Step-by-step solved example

a = ⟨2, 0, 0⟩ m, b = ⟨0, 3, 0⟩ m. Find |a×b| and the direction.

Magnitude |a×b| is the parallelogram area.

1. Magnitude
|a||b|sin90° = 2×3×1 = 6 m².
$∣ a \times b ∣ = 6$
2. Direction
i × j = k, so a×b = ⟨0, 0, 6⟩.
$⟨ 0, 0, 6 ⟩$

Answer: |a×b| = 6 m² along +z

10 practice questions

0/10 correct

1.|a| = 4, |b| = 3, θ = 90°. |a×b| =

2.a × a equals

3.i × j equals

4.Area of the parallelogram spanned by a and b is

5.a×b = −b×a means the cross product is

6.|a| = 2, |b| = 2, θ = 30°. |a×b| =

7.Parallel vectors have |a×b| equal to

8.The direction of a×b is

9.Triangle area from two sides a, b sharing a vertex is

10.|⟨2, 0, 0⟩ × ⟨0, 5, 0⟩| equals