If vector(a) and vector(b) are two vectors such that |vector(a + b)| = |vector a|

Question

If \(\vec{a}\) and \(\vec{b}\) are two vectors such that \(\big|\vec{a} + \vec{b} \big|=\big|\vec{a} \big|,\) then prove that vector \(2\vec{a}+\vec{b}\) is perpendicular to vector \(\vec{b}.\)

Answer 1

\(\because \big|\vec{a} + \vec{b} \big|=\big|\vec{a} \big|\)

\((2\vec{a}+\vec{b})\) is perpendicular to \(\vec{b}.\)