First slide

Scalar triple product of vectors

Question

If $a (\vec{α} \times \vec{β}) + b (\vec{β} \times \vec{γ}) + c (\vec{γ} \times \vec{α}) = 0$ and at least one of a, b and c is non-zero, then vectors $\vec{α}, \vec{β} and \vec{γ}$ are

Moderate

A

parallel
B

co-planar
C

mutually perpendicular
D

none of these

Solution

Taking dot Product of $a (\vec{α} \times \vec{β}) + b (\vec{β} \times \vec{γ}) + c (\vec{γ} \times \vec{α}) = 0$ with $\vec{α}, \vec{β} and \vec{γ}$ respectively
we have,
$a [\vec{α} \vec{β} \vec{γ}] = 0$
$\begin{aligned} b [\vec{α} \vec{β} \vec{γ}] = 0 \\ c [\vec{α} \vec{β} \vec{γ}] = 0 \end{aligned}$
Since at least one of a, b and c $\neq$ 0, we have
$[\vec{α} \vec{β} \vec{γ}] = 0$
Hence, $\vec{α}, \vec{β} and \vec{γ}$ are co-planar.

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