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If $\mathbf{a}=\hat{\mathbf{i}}+\hat{\mathbf{j}}$ and $\mathbf{b}=3 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}$, then the vector $\mathbf{r}$ satisfying the equations $\mathbf{r} \times \mathbf{a}=\mathbf{b} \times \mathbf{a}$ and $\mathbf{r} \times \mathbf{b}=\mathbf{a} \times \mathbf{b}$ is
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2355 Upvotes
Verified Answer
The correct answer is:
$4 \hat{\mathbf{i}}-\hat{\mathbf{j}}$
We have,
$\begin{aligned}
& r \times \mathbf{a}=\mathbf{b} \times \mathbf{a} \text { and } \mathbf{r} \times \mathbf{b}=\mathbf{a} \times \mathbf{b} \\
& \Rightarrow \mathbf{r} \times \mathbf{a}=-(\mathbf{r} \times \mathbf{b}) \\
& \Rightarrow \mathbf{r} \times(\mathbf{a}+\mathbf{b})=0 \\
& \therefore \mathbf{r} \text { is paralleI to } \mathbf{a}+\mathbf{b} \\
& \Rightarrow \mathbf{r}=\mathbf{a}+\mathbf{b}=(\hat{\mathrm{i}}+\hat{\mathrm{j}})+(3 \hat{\mathrm{i}}-2 \hat{\mathrm{j}})=4 \hat{\mathbf{i}}-\hat{\mathrm{j}}
\end{aligned}$
$\begin{aligned}
& r \times \mathbf{a}=\mathbf{b} \times \mathbf{a} \text { and } \mathbf{r} \times \mathbf{b}=\mathbf{a} \times \mathbf{b} \\
& \Rightarrow \mathbf{r} \times \mathbf{a}=-(\mathbf{r} \times \mathbf{b}) \\
& \Rightarrow \mathbf{r} \times(\mathbf{a}+\mathbf{b})=0 \\
& \therefore \mathbf{r} \text { is paralleI to } \mathbf{a}+\mathbf{b} \\
& \Rightarrow \mathbf{r}=\mathbf{a}+\mathbf{b}=(\hat{\mathrm{i}}+\hat{\mathrm{j}})+(3 \hat{\mathrm{i}}-2 \hat{\mathrm{j}})=4 \hat{\mathbf{i}}-\hat{\mathrm{j}}
\end{aligned}$
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