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How does the cross product of two vectors relate to the area of a parallelogram?

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vector parallelogram for addition and subtraction — linear algebra One method of adding and subtracting vectors is to place their tails together and then supply two more sides to form a parallelogram. The vector from their tails to the opposite corner of the parallelogram is equal to the sum of the original vectors. The vector between their heads (starting from the vector being subtracted) is equal to their difference.

In linear algebra and vector analysis, the cross product of two vectors, $𝐚$ and $𝐛$ , is a vector whose magnitude is directly related to the geometry of the parallelogram they form when placed tail-to-tail.12

The magnitude of the cross product is defined by the formula:
$| 𝐚 \times 𝐛 | = | 𝐚 | | 𝐛 | \sin θ$
where $| 𝐚 |$ and $| 𝐛 |$ are the magnitudes of the vectors and $θ$ is the angle between them.1

Geometrically, if you treat $| 𝐚 |$ as the base of the parallelogram, then $| 𝐛 | \sin θ$ represents its perpendicular height. Consequently, the magnitude of the cross product, $| 𝐚 \times 𝐛 |$ , is exactly equal to the area of the parallelogram formed by these two vectors. This relationship holds because the area of any parallelogram is calculated as the product of its base and its height.12