Cross product → a perpendicular vector: The dot product of two vectors is a number. The cross product v×w is a vector instead — and it points at right angles to both v and w (straight out of the flat sheet they lie in).
You get it component by component with the rule below. A quick check: the answer must be perpendicular to both originals, so v·(v×w) should come out 0.
IB-style question — find a perpendicular direction
Two struts of a 3D frame have direction vectors v = (2, 1, 3) and w = (1, 4, 2).
Find a vector perpendicular to both struts.
Step by step
- A vector perpendicular to both is v×w. Take each component with the rule.
- Work out each bracket.
- Check it is perpendicular: dot it with v (should be 0).
Final answer
(−10, −1, 7) is perpendicular to both struts. (Any scalar multiple, e.g. (10, 1, −7), works too.)
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Its length is an area: The length of the cross product is the area of the parallelogram with sides v and w:
|v×w| = |v||w| sin θ.
A triangle is half a parallelogram, so the area of triangle ABC is ½|AB×AC| — two sides drawn from the same corner, crossed, halved.
IB-style question — area of a 3D triangle
A triangular solar panel has corners A(1, 0, 2), B(3, 1, 2) and C(2, 2, 4) (metres).
Find its area.
Step by step
- Make two side vectors from the same corner A.
- Cross them with the component rule.
- Find its length.
- The triangle is half the parallelogram.
Final answer
The panel's area is ½√29 ≈ 2.69 m².