First slide

Vector triple product

Question

If $\vec{α} ∥ (\vec{β} \times \vec{γ}), then (\vec{α} \times \vec{β}) \cdot (\vec{α} \times \vec{γ})$ equals to

Moderate

A

$| \vec{α} |^{2} (\vec{β} \cdot \vec{γ})$
B

$| \vec{β} |^{2} (\vec{γ} \cdot \vec{α})$
C

$| \vec{γ} |^{2} (\vec{α} \cdot \vec{β})$
D

$| \vec{α} | | \vec{β} | | \vec{γ} |$

Solution

$\begin{array}{l} \vec{α} ∥ (\vec{β} \times \vec{γ}) \Rightarrow \vec{α} ⊥ \vec{β} and \vec{α} ⊥ \vec{γ} \\ Now, (\vec{α} \times \vec{β}) \cdot (\vec{α} \times \vec{γ}) \\ = | \vec{α} |^{2} (\vec{β} \cdot \vec{γ}) - (\vec{α} \cdot \vec{β}) (\vec{α} \cdot \vec{γ}) \\ = | \vec{α} |^{2} \cdot (\vec{β} \cdot \vec{γ}) \end{array}$

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