Q: It is found that $\displaystyle |\vec{A} + \vec{B} | = |\vec{A}| $ . This necessarily implies.

(a) $\displaystyle \vec{B} = 0 $

(b) $\displaystyle \vec{A} ,\vec{B} $ are antiparallel

(c) $\displaystyle \vec{A} ,\vec{B} $ are perpendicular

(d) $\displaystyle \vec{A} ,\vec{B} $ ≤ 0

Ans: (a) , (b)

Sol: $\displaystyle |\vec{A} + \vec{B} |^2 = |\vec{A}|^2 $

$\displaystyle (\vec{A} + \vec{B} ).(\vec{A} + \vec{B} ) = \vec{A}.\vec{A} $

$\displaystyle A^2 + B^2 + 2 \vec{A}.\vec{B} = A^2 $

$\displaystyle B^2 + 2 \vec{A}.\vec{B} = 0 $