Question

Let $ \overrightarrow{b} = 4\hat{i} +3\hat{j} $ and $ \overrightarrow{c} $ be two vectors perpendicular to each other $ xy-plane $ , then a vector in the same plane having projections $ 1 $ and $ 2 $ along $ \overrightarrow{b} \,and\, \overrightarrow{c} $ respectively, is

• A. $ \hat{ i } + 2\hat{ j } $
• B. $ 2\hat{ i } - \hat{ j } $
• C. $ 2\hat{ i } + \hat{ j } $
• D. $None\, of\, these$

Answer 1

Answer: (B)

Let $\vec{c}=x \hat{i}+y \hat{j}$, then
$\vec{b} \bot \vec{c} \Rightarrow \vec{b}\cdot\vec{c} =4x+3y=0$
$\Rightarrow \frac{x}{3}=\frac{y}{-4}=\lambda$
$\Rightarrow x=3\lambda$,
$y=-4\lambda$
$\therefore \vec{c}=\lambda \left(3\hat{i}-4\hat{j}\right)$
Let the required vector be $\vec{a}=a_{1} \hat{i}+a_{2}\, \hat{j}$, then the projections of $\vec{a}$ on $\vec{b}$ and $\vec{c}$ are $\frac{\vec{a}\cdot\vec{b}}{\left|\vec{b}\right|}$ and $\frac{\vec{a}\cdot\vec{c}}{\left|\vec{c}\right|}$ respectively
$\therefore \frac{\vec{a}\cdot\vec{b}}{\left|\vec{b}\right|}=1$ and $\frac{\vec{a}\cdot\vec{c}}{\left|\vec{c}\right|}=2$ (given)
$\Rightarrow 4a_{1}+3a_{2}=5 $
and $3a_{1}-4a_{2}=10 $
$\Rightarrow a_{1}=2, a_{2}=-1 $
Hence, the required vector $=2 \hat{i}-\hat{j}$