Question

Show the vectors\(u\)and\(v\)has same length, when\({\rm{u}} + {\rm{v}}\) and\({\rm{u - v}}\)are orthogonal.

Short answer

The answer is stated below.

Step-by-step solution

Formula of condition for orthogonal.

Two vectors\({\rm{u}}\)and\({\rm{v}}\)are orthogonal if and only if\({\rm{u}} \cdot {\rm{v}} = 0.\)

Use the condition for orthogonal for further calculation.

Find the dot product between\((u + v) \cdot (u - v)\).

Apply the condition for orthogonal, that is\((u + v) \cdot (u - v) = 0.\)

\(\begin{aligned}{l}|u{|^2} - |v{|^2} &= 0\\|u{|^2} &= |v{|^2}\end{aligned}\)

Take square roots on both sides.

\(\begin{aligned}{l}\sqrt {|u{|^2}} &= \sqrt {|v{|^2}} \\|u| &= |v|\end{aligned}\)

Therefore, the length of\(u\) and\(v\) is same, id\({\rm{u}} + {\rm{v}}\) and\({\rm{u}} - {\rm{v}}\) are orthogonal.