Question

When will A cross B be equal to B cross A?​

Answer 1

vector product of two vectors is not commutative that is A cross B not equal to B cross A .

In this case magnitudes are equal but directions are opposite. It can be A cross B = -B cross A.

Answer 2

Question: When will $\vec A \times \vec B$ be equal to $\vec B \times \vec A$.

Answer:

$\vec A \times \vec B$ = $\vec B \times \vec A$ holds only when the angle between the vectors A and B is $\pi$.

Step-by-step explanation:

Cross product: The cross product or vector product is a binary operation on two vectors in a three-dimensional space.

Since the cross product is anti-commutative, i.e.,

$\vec A \times \vec B=-\vec B \times \vec A$

Given: $\vec A \times \vec B=\vec B \times \vec A$

We know that,

$\vec A \times \vec B=ABsin\theta \hat{n}$ . . . . . (1)

where $\theta$ is the angle between the vectors A and B.

Similarly,

$\vec B \times \vec A=BAsin(2\pi -\theta) \hat{n}$ . . . . (2)

          $=-BAsin\theta \hat n$ (Since $sin(2\pi -\theta)=-sin\theta$)

From (1) and (2), we have

⇒ $sin(2\pi-\theta)=sin\theta$

⇒ $2\pi-\theta=\theta$

⇒ $2\pi=2\theta$

⇒ $\theta =\pi$

Thus, the angle between the vectors A and B must be $\pi$.

Or, Vectors A and B must be in opposite direction.

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