First slide

Scalar triple product of vectors

Question

$\vec{a} and \vec{b}$ are two unit vectors that are mutually perpendicular. A unit vector that is equally inclined to $\vec{a}, \vec{b} and \vec{a} \times \vec{b} is$

Moderate

A

$\frac{1}{\sqrt{2}} (\vec{a} + \vec{b} + \vec{a} \times \vec{b})$
B

$\frac{1}{2} (\vec{a} \times \vec{b} + \vec{a} + \vec{b})$
C

$\frac{1}{\sqrt{3}} (\vec{a} + \vec{b} + \vec{a} \times \vec{b})$
D

$\frac{1}{3} (\vec{a} + \vec{b} + \vec{a} \times \vec{b})$

Solution

Let the required vector $\vec{r}$ be such that $\vec{r} = x_{1} \vec{a} + x_{2} \vec{b} + x_{3} \vec{a} \times \vec{b}$
We must have $\vec{r} \cdot \vec{a} = \vec{r} \cdot \vec{b} = \vec{r} \cdot (\vec{a} \times \vec{b})$ (as $\vec{r}, \vec{a}, \vec{b} and \vec{a} \times \vec{b}$ are unit vectors and $\vec{r}$ is equally inclined to $\vec{a}, \vec{b} and \vec{a} \times \vec{b}$ )
$\begin{array}{l} now \vec{r} \cdot \vec{a} = x_{1}, \vec{r} \cdot \vec{b} = x_{2}, \vec{r} \cdot (\vec{a} \times \vec{b}) = x_{3} \\ \Rightarrow \vec{r} = λ (\vec{a} + \vec{b} + (\vec{a} \times \vec{b})) \\ also, \vec{r} \cdot \vec{r} = 1 \\ \Rightarrow λ^{2} (\vec{a} + \vec{b} + \vec{a} \times \vec{b}) \cdot (\vec{a} + \vec{b} + (\vec{a} \times \vec{b})) = 1 \\ or λ^{2} (| \vec{a} |^{2} + | \vec{b} |^{2} + | \vec{a} \times \vec{b} |^{2}) = 1 \\ or λ^{2} = \frac{1}{3} \\ or λ = \pm \frac{1}{\sqrt{3}} \end{array}$
$\Rightarrow \vec{r} = \pm \frac{1}{\sqrt{3}} (\vec{a} + \vec{b} + \vec{a} \times \vec{b})$

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