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Multiplication with Vectors

For two parti...

Question

For two particular vectors $\to A$ and $\to B$ it is known that $\to A \times \to B = \to B \times \to A .$ What must be true about the two vectors ?

A

At least one of the two vectors must be the zero vector.

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B

\to A \times \to B = \to B \times \to A

is true for any two vectors.

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C

One of the two vectors is a scalar multiple of the other vector.

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D

The two vectors must be perpendicular to each other.

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Solution

The correct option is C One of the two vectors is a scalar multiple of the other vector.
Given,
$\to A \times \to B = \to B \times \to A$
But from vector product, $\to A \times \to B = - (\to B \times \to A)$
$\Rightarrow \to A \times \to B = - (\to A \times \to B)$
$\Rightarrow 2 (\to A \times \to B) = 0$
$\Rightarrow \to A \times \to B = 0$
$\Rightarrow \to A ∥ \to B$
$\Rightarrow \to A = c \to B$ where c is a scalar

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