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If $\mathbf{p}=\hat{\mathbf{i}}+\hat{\mathbf{j}}, \mathbf{q}=4 \hat{\mathbf{k}}-\hat{\mathbf{j}}$ and $\mathbf{r}=\hat{\mathbf{i}}+\hat{\mathbf{k}}$, then the unit vector in tile direction of $3 p+q-2 r$ is
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Verified Answer
The correct answer is:
$\frac{1}{3}(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+2 \hat{\mathbf{k}})$
We have, $\mathbf{p}=\hat{\mathbf{i}}+\hat{\mathbf{j}}, \mathbf{q}=4 \hat{\mathbf{k}}-\hat{\mathbf{j}}$ and $\mathbf{r}=\hat{\mathbf{i}}+\hat{\mathbf{k}}$
$\begin{aligned}
\text { So, } 3 \mathbf{p}+\mathbf{q}-2 \mathbf{i} &=3 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}-\hat{\mathfrak{j}}-2 \hat{\mathbf{i}}-2 \hat{\mathbf{k}} \\
&=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}
\end{aligned}$
Now, required unit vector $=\frac{\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}}{\sqrt{1+4+4}}$
$=\frac{1}{3}(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+2 \hat{\mathbf{k}})$
$\begin{aligned}
\text { So, } 3 \mathbf{p}+\mathbf{q}-2 \mathbf{i} &=3 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}-\hat{\mathfrak{j}}-2 \hat{\mathbf{i}}-2 \hat{\mathbf{k}} \\
&=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}
\end{aligned}$
Now, required unit vector $=\frac{\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}}{\sqrt{1+4+4}}$
$=\frac{1}{3}(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+2 \hat{\mathbf{k}})$
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