Question

Which of the following is not a unit vector for all values of $\theta$?

• A. $\left(\cos \theta \right)i-\left(\sin \theta \right)j$
• B. $\left(\sin \theta \right)i+\left(\cos \theta \right)j$
• C. $\left(\sin 2\theta \right)i-\left(\cos \theta \right)j$
• D. $\left(\cos 2\theta \right)i-\left(\sin 2\theta \right)j$

Answer 1

The correct option is C $\left(\sin 2\theta \right)i-\left(\cos \theta \right)j$

Let $n$ be any vector.

Then $n$ is a unit vector, if $||n||=1$

We know that, ${\cos }^2\theta +{\sin }^2\theta =1$

For $n=\left(\sin 2\theta \right)i-\left(\cos \theta \right)j$

$||n||={\sin }^{2}2\theta +{\cos }^2\theta \ne 1$

Hence, $\left({\sin }^{2}2\theta \right)i+\left({\cos }^2\theta \right)j$ is not a unit vector.