If |A+B|=|A|+|B| then the angle between the vectors A and B is *0 *180 *90 ?

Apr 25, 2017

${\left\mid m a t h b f A + m a t h b f B \right\mid}^{2} = \left(m a t h b f A + m a t h b f B\right) \cdot \left(m a t h b f A + m a t h b f B\right)$

$= | m a t h b f A {|}^{2} + | m a t h b f B {|}^{2} + m a t h b f A \cdot m a t h b f B + m a t h b f B \cdot m a t h b f A$

As scalar product commutes:

$\implies {\left\mid m a t h b f A + m a t h b f B \right\mid}^{2} = | m a t h b f A {|}^{2} + | m a t h b f B {|}^{2} + 2 | m a t h b f A | | m a t h b f B | \cos \alpha q \quad \square$

If $| m a t h b f A + m a t h b f B | = | m a t h b f A | + | m a t h b f B |$

Then also by squaring each side:

$| m a t h b f A + m a t h b f B {|}^{2} = | m a t h b f A {|}^{2} + | m a t h b f B {|}^{2} + 2 | m a t h b f A | | m a t h b f B | q \quad \triangle$

$\square = \triangle \implies \cos \alpha = 1 , \alpha = 0$