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. $Noneofthese$

Answer 1

Answer: (B)

Let $\vec{c}=x\hat{i}+y\hat{j}$, then
$\vec{b}\perp \vec{c}\Rightarrow \vec{b}\cdot \vec{c}=4x+3y=0$
$\Rightarrow \frac{x}{3}=\frac{y}{-4}=\lambda$
$\Rightarrow x=3\lambda$,
$y=-4\lambda$
$\therefore \vec{c}=\lambda \left(3\hat{i}-4\hat{j}\right)$
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 $\frac{\vec{a}\cdot \vec{b}}{|\vec{b}|}$ and $\frac{\vec{a}\cdot \vec{c}}{|\vec{c}|}$ respectively
$\therefore \frac{\vec{a}\cdot \vec{b}}{|\vec{b}|}=1$ and $\frac{\vec{a}\cdot \vec{c}}{|\vec{c}|}=2$ (given)
$\Rightarrow 4a_1+3a_2=5$
and $3a_1-4a_2=10$
$\Rightarrow a_1=2,a_2=-1$
Hence, the required vector $=2\hat{i}-\hat{j}$