Question

The sum and difference of two perpendicular vectors of equal length are

• A. Perpendicular to each other and of equal length
• B. Perpendicular to each other and of different lengths
• C. Of equal length and have an obtuse angle between them
• D. Of equal length and have an acute angle between them

Answer 1

The correct option is A Perpendicular to each other and of equal length

Consider two $\perp$ vectors of equal magnitude.

Sum $=\overrightarrow{a}+\overrightarrow{b}|\overrightarrow{a}+\overrightarrow{b}|=\sqrt{a^2+b^2}$

Difference $=\overrightarrow{a}-\overrightarrow{b}|\overrightarrow{a}-\overrightarrow{b}|=\sqrt{a^2+b^2}$

Let be angle b/w the vectors $\cos \theta =\frac{(\overrightarrow{a}+\overrightarrow{b})(\overrightarrow{a}-\overrightarrow{b})}{|a^2+b^2||a^2-b^2|}=\frac{a^2-b^2}{a^2+b^2}=0$

$\Rightarrow \theta ={90}^0$.