​​​​​ If vec a and vec b are two vectors such that |vec a + vec b| = |vec b|, then prove that (vec a + 2vec b) is perpendicular to vec a. ​

Question

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

Answer 1

\(|\vec a + \vec b| = |\vec b|\)

⇒ \(|\vec a + \vec b| ^2= |\vec b |^2\)

⇒ \((\vec a + \vec b ). (\vec a + \vec b)= \vec b . \vec b\)

⇒ \(|\vec a |^2 + 2 \vec a .\vec b + |\vec b|^2 = |\vec b|^2\)

⇒ \(|\vec a|^2 + 2 \vec a . \vec b = 0\)

⇒ \(\vec a . \vec a + 2\vec a. \vec b = 0\)

⇒ \(\vec a. (\vec a + 2 \vec b) =0\)

Hence, \((\vec a + 2\vec b) \) is perpendicular to \(\vec a.\)