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

• A

  $\hat{\mathrm{i}}-\hat{\mathrm{j}}-2 \hat{\mathrm{k}}$
• B

  $3 \hat{i}+3 \hat{j}+2 \hat{k}$
• Correct answer
  C

  $-\hat{\mathrm{i}}+7 \hat{\mathrm{j}}+10 \hat{\mathrm{k}}$
• D

  $4 \hat{\mathrm{i}}+5 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}$

Solution

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 $\vec{x}$ $\begin{aligned} & \Rightarrow \vec{b}+\vec{x}=\vec{a} \Rightarrow \vec{x}=\vec{a}-\vec{b} \\ & \Rightarrow \vec{x}=(\hat{i}+3 \hat{j}+13 \hat{k})-(2 \hat{i}-4 \hat{j}+3 \hat{k}) \\ & \Rightarrow \vec{x}=-\hat{i}+7 \hat{j}+10 \hat{k} \end{aligned}$

Hence, the correct answer is option: 'C'.