Question

$\overline{x},\overline{y},\overline{z}$ are non-zero vectors, if $(\overline{x}.\overline{y})=\frac{\pi }{2}$ and $\overline{z}=\overline{x}+\overline{y}$, then

• A. $|\overline{x}{|}^2+|\overline{y}{|}^2+|\overline{z}{|}^2=3$
• B. $|\overline{x}{|}^2+|\overline{y}{|}^2+|\overline{z}{|}^2=1$
• C. $|\overline{x}{|}^2-|\overline{y}{|}^2=|\overline{z}{|}^2$
• D. $|\overline{x}{|}^2+|\overline{y}{|}^2=|\overline{z}{|}^2$

Answer 1

The correct option is D $|\overline{x}{|}^2+|\overline{y}{|}^2=|\overline{z}{|}^2$

$\overrightarrow{x}.\overrightarrow{y}=xy\cos \theta =xy\cos \frac{\pi }{2}=0$

$|\overrightarrow{z}{|}^2=|\overrightarrow{x}+\overrightarrow{y}{|}^2$

$⟹z^2=x^2+y^2+2xy\cos \theta =x^2+y^2$

$⟹|\overrightarrow{x}{|}^2+|\overrightarrow{y}{|}^2=|\overrightarrow{z}{|}^2$