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If $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+3 \hat{\mathrm{j}}+13 \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{b}}=2 \hat{\mathrm{i}}-4 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}$ are two vectors, then the component vector of $\vec{a}$ perpendicular to $\vec{b}$ is
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Verified Answer
The correct answer is:
$-\hat{\mathrm{i}}+7 \hat{\mathrm{j}}+10 \hat{\mathrm{k}}$
Given $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+3 \hat{\mathrm{j}}+13 \hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=2 \hat{\mathrm{i}}-4 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}$
Let the component of vector $\vec{a}$ perpendicular to $\vec{b}$ is $\vec{x}$
$\begin{aligned}
& \Rightarrow \vec{b}+\vec{x}=\vec{a} \Rightarrow \vec{x}=\vec{a}-\vec{b} \\
& \Rightarrow \vec{x}=(\hat{i}+3 \hat{j}+13 \hat{k})-(2 \hat{i}-4 \hat{j}+3 \hat{k}) \\
& \Rightarrow \vec{x}=-\hat{i}+7 \hat{j}+10 \hat{k}
\end{aligned}$
Let the component of vector $\vec{a}$ perpendicular to $\vec{b}$ is $\vec{x}$
$\begin{aligned}
& \Rightarrow \vec{b}+\vec{x}=\vec{a} \Rightarrow \vec{x}=\vec{a}-\vec{b} \\
& \Rightarrow \vec{x}=(\hat{i}+3 \hat{j}+13 \hat{k})-(2 \hat{i}-4 \hat{j}+3 \hat{k}) \\
& \Rightarrow \vec{x}=-\hat{i}+7 \hat{j}+10 \hat{k}
\end{aligned}$
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