Question

If two adjacent sides of a square are represented by the vectors $x\overrightarrow{i}+\overrightarrow{j}+4\overrightarrow{k}$ and $3\overrightarrow{i}+y\overrightarrow{j}$ then $xy=$

• A. $1$
• B. $2$
• C. $3$
• D. $-3$

Answer 1

The correct option is D $-3$

$x\hat{i}+\hat{j}+4\hat{k}$ and $3\hat{i}+y\hat{j}$ are sides of square

$\Rightarrow |x\hat{i}+\hat{j}+4\hat{k}|=|3\hat{i}+y\hat{j}|$

$\Rightarrow x^2+1+16=9+y^2$

$\Rightarrow x^2+8=y^2$ (1)

also $x\hat{i}+\hat{j}+4\hat{k}$ and $3\hat{i}+y\hat{j}$ are perpendicular

$\Rightarrow (x\hat{i}+\hat{j}+4\hat{k}).(3\hat{i}+y\hat{j})=0$

$3x+y=0$

$\Rightarrow y=-3x$ (2)

put (2) in (1)

$x^2+8=9x^2$

$8x^2=8$

$x=\pm 1$

if $x=1$

$y=-3x=-3$

$\Rightarrow xy=-3$

if $x=-1$

$y=-3x=3$

$\Rightarrow xy=-3$

$\therefore xy=-3$