The two vectors $\overrightarrow{\mathrm{A}}=-2\hat{i}+3\hat{\mathrm{j}}+y\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{B}}=\hat{i}+2\hat{\mathrm{j}}+x\hat{\mathrm{k}}$ are perpendicular. Given $x+y=0$. Find x and y?

 $x+y=0$

when the vectors are perpendicular.

$\overrightarrow{\text{A}}\text{.}\overrightarrow{\text{B}}{\text{=A}}_{\text{x}}{\text{B}}_{\text{x}}{\text{+A}}_{\text{y}}{\text{B}}_{\text{y}}{\text{+A}}_{\text{z}}{\text{B}}_{\text{z}}\text{ =0}$

$\overrightarrow{\text{A}}\text{.}\overrightarrow{\text{B}}\text{=-2+6+}xy\text{=0}$

$\Rightarrow xy=-4$

${\left(x-y\right)}^2={\left(x+y\right)}^2-4xy$

$x-y=\pm 4$

$x=2 and y=-2$

or $x=-2 and y=2$