First slide

Scalar triple product of vectors

Question

If $\vec{a} and \vec{b}$ are unit vectors such that $(\vec{a} + \vec{b}) \cdot (2 \vec{a} + 3 \vec{b}) \times (3 \vec{a} - 2 \vec{b}) = \vec{0}$ angle between $\vec{a} and \vec{b}$ is

Moderate

A

0
B

$\frac{π}{2}$
C

$π$
D

indeterminate

Solution

$\begin{matrix} \vec{0} = (\vec{a} + \vec{b}) \cdot (2 \vec{a} + 3 \vec{b}) \times (3 \vec{a} - 2 \vec{b}) \\ = (\vec{a} + \vec{b}) \cdot (- 4 \vec{a} \times \vec{b} - 9 \vec{a} \times \vec{b}) \\ = - 13 (\vec{a} + \vec{b}) \cdot (\vec{a} \times \vec{b}) \end{matrix}$
which is true for all values of $\vec{a} and \vec{b}$

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