VECTOR ALGEBRA — Maths STD 12 Science — Question

2. Unit vector

Solution:

Let $\vec{\text{a}}=(\cos\text{a}\cos\beta)\hat{\text{i}}+(\cos\text{a}\sin\beta)\hat{\text{j}}+(\sin\text{a})\hat{\text{k}}$

$|\vec{\text{a}}|=\sqrt{\cos^2\text{a}\cos^2\beta+\cos^2\text{a}\sin^2\beta+\sin^2\text{a}}$

$=\sqrt{\cos^2\text{a}(\cos^2\beta+\sin^2\beta)+\sin^2\text{a}}$

$=\sqrt{\cos^2\text{a}(1)+\sin^2\text{a}}$

$=\sqrt{\cos^2\text{a}+\sin^2\text{a}}$

$=\sqrt{1}$

$=1$

So, $\vec{\text{a}}$ is a unit vector.