Question

If a and b are vectors such that $|a+b|=|a-b|$ then the angle between a and b is

• A. 60${}^{\circ }$
• B. 120${}^{\circ }$
• C. 30${}^{\circ }$
• D. 90${}^{\circ }$

Answer 1

Answer: (D)

We have, $|a+b|=|a-b|$
On squaring both sides, we get
$|a+b{|}^2=|a-b{|}^2$
$\Rightarrow |\vec{a}{|}^2+|\vec{b}{|}^2+2\vec{a}.\vec{b}=|\vec{a}{|}^2+|\vec{b}{|}^2-2\vec{a}.\vec{b}$
$\Rightarrow 4\vec{a}.\vec{b}=0$
$\Rightarrow \vec{a}.\vec{b}=0$
$\Rightarrow \vec{a}$ and $\vec{b}$ are perpendicular to each other.
So, angle between them is $90^{\circ }$.
Alternative
$\because |a+b|=|a-b|$
$\therefore$ a and b are perpendicular to each other.
So, angle between $a$ and b is $90^{\circ }$