The magnitude of the cross product of two vectors a and b is given by the formula:
|a x b| = |a| |b| sin(θ)
where:
- |a| is the magnitude (length) of vector a,
- |b| is the magnitude of vector b, and
- θ is the angle between the two vectors.
This formula highlights that the magnitude of the cross product depends not only on the lengths of the vectors but also on the sine of the angle between them. The sine function reaches its maximum value (1) when the vectors are perpendicular (θ = 90°), which results in the largest possible magnitude for the cross product. Conversely, if the two vectors are parallel (θ = 0° or θ = 180°), the cross product’s magnitude becomes zero.
In summary, the magnitude of the cross product captures both the size of the individual vectors and their orientation relative to each other, providing insight into the geometric relationship between them.