Dot product vs. cross product: what each tells us
The dot product and cross product both use two vectors, but they give us different information. The dot product relates their directions through cosine. In three dimensions, the cross product gives a perpendicular vector whose length depends on sine.
Use cosine to understand the dot product
For two nonzero vectors of fixed lengths, the dot product is positive when the angle is acute, zero when it is a right angle, and negative when it is obtuse. The largest value occurs when they point the same way, and the smallest when they point in opposite directions.
Use sine to understand the length of the cross product
For fixed lengths, the cross product has its greatest magnitude when the vectors are perpendicular. Its magnitude is zero when the vectors point in the same or opposite directions. The right-hand rule gives the direction of a nonzero cross product.
| Angle | Dot product | Cross product magnitude |
|---|---|---|
| Product of the lengths | ||
| Product of the lengths | ||
| Negative product of the lengths |
Work through an opposite-direction example
Write the planar vectors as and . Their dot product is , and their cross product is . The negative dot product and zero cross product agree with the vectors pointing in opposite directions.