Mathematics › Vector Algebra › Vector Triple Product
If $\mathrm{a}, \mathrm{b}$ and $\mathrm{c}$ are non-zero vectors such that $\mathrm{a}$ and $\mathrm{b}$…
If $\mathrm{a}, \mathrm{b}$ and $\mathrm{c}$ are non-zero vectors such that $\mathrm{a}$ and $\mathrm{b}$ are not perpendicular to each other, then the vector $\mathbf{r}$ which is perpendicular to $\mathbf{a}$ and satisfying $\mathrm{r} \times \mathrm{b}=\mathrm{c} \times \mathrm{b}$ is
$\frac{(\mathbf{a} \times \mathbf{b}) \times \mathbf{c}}{\mathbf{c} \cdot \mathbf{a}}$ $\frac{\mathbf{b} \times(\mathbf{a} \times \mathbf{c})}{\mathbf{b} \cdot \mathbf{c}}$ $\frac{(\mathbf{b} \times \mathbf{c}) \times \mathbf{a}}{\mathbf{a} \cdot \mathbf{b}}$ $\frac{(\mathbf{c} \times \mathbf{b}) \times \mathbf{a}}{\mathbf{a} \cdot \mathbf{c}}$ Solution
Given that, $a, b$ and $c$ are non-zero vectors such that a $\& \mathbf{b}$ are not perpendicular.
Also given, $\mathbf{r} \times \mathbf{b}=\mathbf{c} \times \mathbf{b}$
$
\Rightarrow \mathbf{a} \times(\mathbf{r} \times \mathbf{b})=\mathbf{a} \times(\mathbf{c} \times \mathbf{b})
$
Since, $\mathbf{r} \perp \mathbf{a}$, therefore $\mathbf{a} . \mathbf{r}=0$
$
\begin{aligned}
& \Rightarrow \quad(\mathbf{a} \cdot \mathbf{b}) \mathbf{r}-(\mathbf{a} \cdot \mathbf{r}) \mathbf{b}=\mathbf{a} \times(\mathbf{c} \times \mathbf{b}) \\
& \Rightarrow \quad(\mathbf{a} \cdot \mathbf{b}) \mathbf{r}=\mathbf{a} \times(\mathbf{c} \times \mathbf{b})
\end{aligned}
$
$\begin{aligned} & \Rightarrow \quad=\frac{\mathbf{a} \times(\mathbf{c} \times \mathbf{b})}{\mathbf{a} \cdot \mathbf{b}}=-\frac{[(\mathbf{c} \times \mathbf{b}) \times \mathbf{a}]}{\mathbf{a} \cdot \mathbf{b}} \\ & =-\frac{[-(\mathbf{b} \times \mathbf{c}) \times \mathbf{a}]}{\mathbf{a} \cdot \mathbf{b}}=\frac{(\mathbf{b} \times \mathbf{c}) \times \mathbf{a}}{\mathbf{a} \cdot \mathbf{b}}\end{aligned}$
Asked in: AP EAMCET 2017 (26 Apr Shift 1)
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