If the cross product of two vectors vanishes, what can you say about their directions?
The correct answer and explanation is:
When the cross product of two vectors vanishes, the vectors must be parallel or anti-parallel to each other. This means that the two vectors are either pointing in the same direction (parallel) or in exactly opposite directions (anti-parallel). The reason for this is that the magnitude of the cross product is given by: ∣A×B∣=∣A∣∣B∣sin(θ)|A \times B| = |A| |B| \sin(\theta)
Where ∣A∣|A| and ∣B∣|B| are the magnitudes of vectors A and B, and θ\theta is the angle between them. If the cross product is zero, then either:
- The angle θ\theta between the two vectors is 0° (which means the vectors are parallel), or
- The angle θ\theta is 180° (which means the vectors are anti-parallel).
In both cases, the sine of these angles is zero, resulting in a cross product of zero.
This condition tells us that the two vectors are not perpendicular to each other, as the cross product of two perpendicular vectors is maximized. Hence, if the cross product is zero, it indicates that the vectors lie along the same line or are scalar multiples of each other.
It is important to note that while parallel and anti-parallel vectors have the same or opposite directions, the vectors could still have different magnitudes. The vanishing of the cross product only tells us about their directionality, not their magnitude. Therefore, for the cross product to vanish, the vectors must be collinear, regardless of their size.