Question

For vectors $\vec A$ and $\vec B$ making an angle $\theta $, which one of the following is correct?
A) $\vec A \times \vec B = \vec B \times \vec A$
B) $\vec A \times \vec B = AB\sin \theta $
C) $\vec A \times \vec B = AB\cos \theta $
D) $\vec A \times \vec B = - \vec B \times \vec A$

Answer 1

Hint: The cross product of two vectors gives us a vector that is perpendicular to both the vectors i.e. perpendicular to the plane containing the two vectors. The magnitude of the cross product is equal to the area of the parallelogram formed by two vectors.

Formula used: $\vec A \times \vec B = AB\vec n\sin \theta $ where $A$ and $B$ are the magnitudes of their respective vectors and $\vec n$ is a unit vector in the direction perpendicular to both $\vec A$ and $\vec B$.

Complete step by step solution:
The cross product of two vectors can be written in the form of $\vec A \times \vec B = AB\hat n\sin \theta $.
In option (A), $\vec A \times \vec B \ne \vec B \times \vec A$ , since cross product isn’t commutative in nature so it is not correct.
In option (B) , the cross product has no direction defined by the unit vector $\hat n$ so it is not correct either.
In option (C), there is no direction defined as well as $\cos \theta $ has been given instead of $\sin \theta $ so it is not correct.
In option (D), the anti-commutative property of cross product is utilized i.e. $\vec A \times \vec B = - \vec B \times \vec A$ which tells us that when switching the order of cross product, the direction of the unit vector will be flipped.

Hence, option (D) is the correct answer.

Additional Information:
To find the direction of the unit vector perpendicular to both $\vec A$ and $\vec B$, we can use the right-hand rule. On placing your index finger in the direction of $\vec A$ and middle finger in the direction of $\vec B$ and raising your thumb above your hand, the direction of the thumb gives us the direction of $\hat n$.

Note:
We must be careful in not getting caught in option (B) since it is almost correct but it misses a unit vector that defines the direction of the cross product. While dot products don’t have a direction, cross products will have a direction vector and hence option (B) must be discarded.