Question

Which of the following is/are a unit vector for all values of $\theta ?$

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

Answer 1

Answer: (A), (B), (D)

$\left(\cos 2\theta \right)\hat{i}-\left(\sin 2\theta \right)\hat{j}$

We know, unit vector has magnitude equal to $1.$

$a:\left(\cos \theta \right)\hat{i}-\left(\sin \theta \right)\hat{j}$
$\text{magnitude}=\sqrt{{\cos }^2\theta +{\sin }^2\theta }=1\forall \theta \in R$
So, a unit vector.

$b:\left(\sin \theta \right)\hat{i}+\left(\cos \theta \right)\hat{j}$
$\text{magnitude}=\sqrt{{\sin }^2\theta +{\cos }^2\theta }=1\forall \theta \in R$
So, a unit vector.

$c:\left(\sin 2\theta \right)\hat{i}-\left(\cos \theta \right)\hat{j}$
$\text{magnitude}=\sqrt{{\sin }^{2}2\theta +{\cos }^2\theta }\ne 1$ for all $\theta \in R$
so, not a unit vector.

$d:\left(\cos 2\theta \right)\hat{i}-\left(\sin 2\theta \right)\hat{j}$
$\text{magnitude}=\sqrt{{\cos }^{2}2\theta +{\sin }^{2}2\theta }=1\forall \theta \in R$
so, a unit vector.