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 3D vectors and returns a new vector perpendicular to both, which is how two edges give an area. The cycle to memorize is i cross j = k, with direction set by the right hand rule. Dotting the result with either input gives zero, which confirms the right angle. The vector k has length 1, so the magnitude of i cross j is 1.
With the determinant formula, (1, 0, 0) cross (0, 1, 0) comes out as (0, 0, 1). Order matters: swapping the inputs negates the result, so j cross i is minus k. In general, v cross u equals minus (u cross v) for any pair.
The length of a cross product equals the area of the parallelogram the inputs span. The vectors (3, 0, 0) and (0, 2, 0) cross to (0, 0, 6), so their parallelogram has area 6. A triangle is half of that: with edges (5, 0, 0) and (0, 2, 0) the cross has 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.
2 · Watch
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.