Question

Let $\overrightarrow{u}=\hat{i}+\hat{j},\overrightarrow{v}=\hat{i}-\hat{j}$ and $\overrightarrow{w}=\hat{i}+2\hat{j}+3\hat{k}$. If $\hat{n}$ is a unit vector $\overrightarrow{u}\cdot \hat{n}=0$ and $\overrightarrow{v}\cdot \hat{n}=0$, then $|\hat{w}\cdot \hat{n}|$ is equal to?

• A. $1$
• B. $2$
• C. $3$
• D. $0$

Answer 1

Answer: (C)

$3$

$\overrightarrow{u}=\overrightarrow{i}+\overrightarrow{j}\overrightarrow{v}=\overrightarrow{i}-\overrightarrow{j}\overrightarrow{w}=\overrightarrow{i}+2\overrightarrow{j}+3\overrightarrow{k}letn=a\hat{i}+b\hat{j}+c\hat{k}$

Given n is the unit vector $\left|n\right|=1a^2+b^2+c^2=1\overrightarrow{u}\hat{n}=0(\hat{i}+\hat{j})(a\hat{i}+b\hat{j}+c\hat{k})=0a+b=0⟶1\overrightarrow{v}\hat{n}=0(\hat{i}-\hat{j})(a\hat{i}+b\hat{j}+c\hat{k})=0a-b=0⟶2$

Solving $1$ and $2$

$a=0,b=0$

We know that,$a^2+b^2+c^2=1c=1\therefore n=1\hat{k}\left|\overrightarrow{w}\hat{n}\right|=((\overrightarrow{i}+2\overrightarrow{j}+3\overrightarrow{k}).\left(\hat{k}\right))=3$