First slide

Vector triple product

Question

If $(\vec{a} \times \vec{b}) \times (\vec{c} \times \vec{d}) \cdot (\vec{a} \times \vec{d}) = 0$ then which of the following may be true?

Moderate

A

$\vec{a}, \vec{b}, \vec{c} and \vec{d}$ are necessarily coplanar
B

$\vec{a}$ lies in the plane of $\vec{c} and \vec{d}$
C

$\vec{b}$ lies in the plane of $\vec{a} and \vec{d}$
D

$\vec{c}$ lies in the plane of $\vec{a} and \vec{d}$

Solution

$\begin{array}{l} (\vec{a} \times \vec{b}) \times (\vec{c} \times \vec{d}) \cdot (\vec{a} \times \vec{d}) = 0 \\ or ([\vec{a} \vec{c} \vec{d}] \vec{b} - [\vec{b} \vec{c} \vec{d}] \vec{a}) \cdot (\vec{a} \times \vec{d}) = 0 \end{array}$
$or [\vec{a} \vec{c} \vec{d}] [\vec{b} \vec{a} \vec{d}] = 0$
Hence, either $\vec{c}$ and $\vec{b}$ must lie in the plane of $\vec{a} and \vec{d}$

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