The Cross Product and Oriented Area
The other product of two vectors returns a vector perpendicular to both, whose length is the area of the parallelogram they span and whose direction records the orientation.
What a learner can do afterwards
- Compute a cross product and check it is perpendicular to both inputs
- Use the magnitude to find the area of a triangle in space
- Show that swapping the order reverses the result
1 · Read
The cross product takes two vectors in space and returns a vector perpendicular to both. For u = (u1, u2, u3) and v = (v1, v2, v3), u cross v = (u2v3 - u3v2, u3v1 - u1v3, u1v2 - u2v1). Each entry is a 2 by 2 determinant. The right hand rule sets the direction: fingers along u, curl them toward v, and the thumb points along u cross v. Dotting the result with u or with v gives 0, which checks the right angle.
For the basis vectors, i cross j = k. Moving around the cycle i, j, k, i in order gives a positive result, so each in-order product is the next vector of the cycle. Swapping the inputs negates every entry, so v cross u = -(u cross v), and products against the cycle come out negative.
The length of u cross v equals the area of the parallelogram the two vectors span. It also equals the length of u times the length of v times the sine of the angle between them. A triangle with the same two edges has half that area. With edges (2, 0, 0) and (0, 4, 3), the cross is (0, -6, 8), of length 10, so the triangle area is 5.
Keep the pair straight: the dot product returns a number tied to the angle, while the cross product returns a vector tied to the oriented area. When you only need an area, take the length and skip worrying about direction.
Perpendicular output, flipped order, and a length that measures area, with triangles taking half.
The lesson and quiz items were written for this dataset. The lesson and quiz items q1, q7 and q8 were later revised.
Extra items x1 to x8 were written for this dataset. The rewritten readings were written for this dataset. Other reading entries summarise the Khan Academy article Cross products. Videos are from Khan Academy.
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