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.

Let us assume two vectors $A$ and $B$ and two vectors are of the same magnitude.

Thus, sum $=\overrightarrow{A}+\overrightarrow{B}=|\overrightarrow{A}+\overrightarrow{B}|=\sqrt{(A^2+B^2)}$

Difference $=\overrightarrow{A}-\overrightarrow{B}=|\overrightarrow{A}-\overrightarrow{B}|=\sqrt{(A^2+B^2)}$

Hence, the angle between two vectors is:

$Cos\theta =\frac{\left[\right(\overrightarrow{A}+\overrightarrow{B}\left)\right(\overrightarrow{A}-\overrightarrow{B}\left)\right]}{\left[\right|A^2+B^2\left|\right|A^2-B^2\left|\right]}=0$

Hence, $\theta =90°$

Therefore, the sum and difference of two perpendicular vectors of equal length are perpendicular to each other and of equal length.