Question

Let $a=4\overrightarrow{i}+3\overrightarrow{j}$ and $b$ be two vectors perpendicular to each other in $xy-$ plane. The vectors $c$ in the same plane having projections $1$ and $2$ along $a$ and $c$ are

• A. $-\frac{2}{3}\overrightarrow{i}+\frac{11}{2}\overrightarrow{j}$
• B. $2\overrightarrow{i}-\overrightarrow{j}$
• C. $-\frac{2}{5}\overrightarrow{i}+\frac{11}{2}\overrightarrow{j}$
• D. $\frac{2}{3}\overrightarrow{i}+\frac{11}{2}\overrightarrow{j}$

Answer 1

Answer: (B), (C)

$-\frac{2}{5}\overrightarrow{i}+\frac{11}{2}\overrightarrow{j}$
$2\overrightarrow{i}-\overrightarrow{j}$

Let $b=x\overrightarrow{i}+y\overrightarrow{j}$

Since $a$ is perpendicular to $b$, so $4x+3y=0$

Thus $b=x(\overrightarrow{i}-\frac{4}{3}\overrightarrow{j})$
Let $c=u\overrightarrow{i}+v\overrightarrow{j}$ be the required vector.

According to the given condition,
$1=\frac{c\cdot a}{\left|a\right|}\Rightarrow 4u+3v=5$

Also $2=\frac{c\cdot b}{\left|b\right|}\Rightarrow \frac{ux-\left(4/3\right)vx}{\sqrt{x^2(1+16/9)}}=2\Rightarrow 3u-4v=\pm 10$
Solving these equations, we have $u=2$ and $v=-1$ or $u=-\frac{2}{5},v=\frac{11}{5}$