Question

Vector $\overrightarrow{a}$ has components $a_x=3,a_y=4.$ The componenets of a vector $\overrightarrow{c}$ which is perpendicular to $\overrightarrow{a}$ and has a magnitude of 5 units will be

• A. $c_x=4,c_y=-3$
• B. $c_x=-4,c_y=3$
• C. $c_x=-4,c_y=-3$
• D. Both (a) and (b)

Answer 1

The correct option is D Both (a) and (b)

$\overrightarrow{a}=3\hat{i}+4\hat{j}$
Let $\overrightarrow{c}=c_x\hat{i}+c_y\hat{j}$
As$\overrightarrow{c}$ is perpendicular to $\overrightarrow{a}$
$\Rightarrow \overrightarrow{c}.\overrightarrow{a}=0$
$\therefore 3c_x+4c_y=0$
$\Rightarrow c_y=-\frac{3}{4}{c}_x$ ...(i)

We know $|\overrightarrow{c}|=5,$
Thus, $c_x^2+c_y^2=25$
$\Rightarrow c_x^2+\frac{9c_x^2}{16}=25$
$\therefore c_x=\pm 4$ and from equation (i), $c_y=\mp 3.$