Question

The vectors $\overrightarrow{b}$ and $\overrightarrow{c}$ are in the direction of north-east and north-west respectively and $|\overrightarrow{b}|=|\overrightarrow{c}|=4$. The magnitude and direction of the vector $\overrightarrow{d}=\overrightarrow{c}-\overrightarrow{b},$ is

• A. $4\sqrt{2}$, towards north
• B. $4\sqrt{2}$, towards west
• C. $4$, towards east
• D. $4$, towards south

Answer 1

The correct option is B

$4\sqrt{2}$, towards west

Clearly, $\overrightarrow{b}\perp \overrightarrow{c},\therefore \overrightarrow{b}\cdot \overrightarrow{c}=0$
Now, $\overrightarrow{d}=\overrightarrow{c}-\overrightarrow{b}$ $\Rightarrow |\overrightarrow{d}{|}^2=|\overrightarrow{c}-\overrightarrow{b}{|}^2$
$=|\overrightarrow{c}{|}^2+|\overrightarrow{b}{|}^2-2\overrightarrow{b}\cdot \overrightarrow{c}=16+16-0$
$\Rightarrow |\overrightarrow{d}|=\sqrt{32}=4\sqrt{2}$
Now the direction of $\overrightarrow{b}$ is north east so the direction of $-\overrightarrow{b}$ will be south west
and since the magnitude of vectors $\overrightarrow{b}$ and $\overrightarrow{c}$ is equal $\overrightarrow{d}$ will lie in the direction of angle bisector of $-\overrightarrow{b}$ and $\overrightarrow{c}$ so the direction of $\overrightarrow{d}$ is west.