Mathematics › Vector Algebra › Vector Triple Product
For any vector $\mathbf{r}$ $\mathbf{i} \times(\mathbf{r} \times \mathbf{i})+\mathbf{j} \times(\mathbf{r}…
For any vector $\mathbf{r}$
$\mathbf{i} \times(\mathbf{r} \times \mathbf{i})+\mathbf{j} \times(\mathbf{r} \times \mathbf{j})+\mathbf{k} \times(\mathbf{r} \times \mathbf{k})$ is equal to
0 $2 \mathrm{r}$ $3 \mathrm{r}$ $4 \mathrm{r}$ Solution
Now, $\mathbf{i} \times(\mathbf{r} \times \mathbf{i})$
$\begin{aligned}
& =(\mathbf{i} \cdot \mathbf{i}) \mathbf{r}-(\mathbf{i} \cdot \mathbf{r}) \mathbf{i} \\
& =\mathbf{r}-r_1 \mathbf{i}
\end{aligned}$
$\begin{aligned}
& \text { Similarly, } \mathbf{j} \times(\mathbf{r} \times \mathbf{j})=\mathbf{r}-r_2 \mathbf{j} \\
& \text { and } \quad \mathbf{k} \times(\mathbf{r} \times \mathbf{k})=\mathbf{r}-r_3 \mathbf{k} \\
& \therefore(\mathbf{i} \times(\mathbf{r} \times \mathbf{i}))+(\mathbf{j} \times(\mathbf{r} \times \mathbf{j}))+(\mathbf{k} \times(\mathbf{r} \times \mathbf{k})) \\
& =3 \mathbf{r}-\left(r_1 \mathbf{i}+r_2 \mathbf{j}+r_3 \mathbf{k}\right) \\
& =3 \mathbf{r}-\mathbf{r}=2 \mathbf{r}
\end{aligned}$
Asked in: AP EAMCET 2011
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