Two non-zero vectors are parallel if and only if their cross product is the zero vector. Therefore, calculate ; if the result is , the vectors are parallel.
For vectors and ,
Two non-zero vectors are parallel if and only if their cross product is the zero vector. Therefore, calculate ; if the result is , the vectors are parallel.
For vectors and ,
where is the angle between them. If both vectors are non-zero, then . Consequently,
This gives or , so the vectors point in the same or opposite direction. In either case, they are parallel.
The test also has a geometric interpretation. The magnitude equals the area of the parallelogram formed by the vectors. Parallel vectors produce a flattened parallelogram with zero area. Conversely, if two non-zero vectors produce zero area, the angle between them has zero sine, so they must lie along the same line.
For example, let
Then
Therefore, and are parallel. This is also confirmed by .
When calculating, write all three components before drawing the conclusion. A single zero component is insufficient: the complete cross product must equal . You may then verify the result by checking whether one vector is a scalar multiple of the other, although this second check is not required.
| Cross-product result | Conclusion |
|---|---|
| , with both vectors non-zero | The vectors are parallel |
| The vectors are not parallel | |
| One vector is | The test gives , but no direction comparison can be made |
A common misconception is that a zero cross product always proves two vectors have the same direction. It also includes opposite directions, and the non-zero-vector condition must be checked.
Exam technique: In Mathematics AA HL, AHL 3.16: Vector product, show the cross-product calculation and explicitly conclude that the vectors are parallel because the result is . This analytic method may be tested on Paper 1 or Paper 2. On Paper 1, keep the arithmetic exact; on Paper 2, technology may check the computation, but you should still show the vector set-up and conclusion.