Find a unit vector in the xy-plane which make | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Let the unit vector in the $xy$-plane be $\vec{u} = \cos 	heta \hat{i} + \sin 	heta \hat{j}$.Given that $\vec{u}$ makes an angle of $45^{\circ}$

Vedclass · Accepted Answer

None of these

Vedclass · Answer

(D) Let the unit vector in the $xy$-plane be $\vec{u} = \cos 	heta \hat{i} + \sin 	heta \hat{j}$.Given that $\vec{u}$ makes an angle of $45^{\circ}$ with $\vec{a} = \hat{i} + \hat{j}$.The angle between $\vec{u}$ and $\vec{a}$ is given by $\cos 45^{\circ} = \frac{\vec{u} \cdot \vec{a}}{|\vec{u}| |\vec{a}|}$.$\frac{1}{\sqrt{2}} = \frac{\cos 	heta + \sin 	heta}{1 \cdot \sqrt{2}}$,which implies $\cos 	heta + \sin 	heta = 1$.Squaring both sides,$1 + 2 \sin 	heta \cos 	heta = 1$,so $\sin 2	heta = 0$,meaning $	heta = 0$ or $	heta = 90^{\circ}$.If $	heta = 0$,$\vec{u} = \hat{i}$. If $	heta = 90^{\circ}$,$\vec{u} = \hat{j}$.Now,check the angle with $\vec{b} = 3\hat{i} - 4\hat{j}$.For $\vec{u} = \hat{i}$,$\cos \phi = \frac{\hat{i} \cdot (3\hat{i} - 4\hat{j})}{1 \cdot 5} = \frac{3}{5} 
eq \cos 60^{\circ} = \frac{1}{2}$.For $\vec{u} = \hat{j}$,$\cos \phi = \frac{\hat{j} \cdot (3\hat{i} - 4\hat{j})}{1 \cdot 5} = \frac{-4}{5} 
eq \frac{1}{2}$.Since neither $\hat{i}$ nor $\hat{j}$ satisfies the second condition,the correct answer is None of these.

Find a unit vector in the $xy$-plane which makes an angle of $45^{\circ}$ with the vector $i + j$ and an angle of $60^{\circ}$ with the vector $3i - 4j$.

Similar Questions

Let $\vec{b}$ and $\vec{c}$ be non-collinear vectors satisfying $\vec{a} \times (\vec{b} \times \vec{c}) + (\vec{a} \cdot \vec{b})\vec{b} = (4 - 2x - \sin y)\vec{b} + (x^2 - 1)\vec{c}$ and $(\vec{c} \cdot \vec{c})\vec{a} = \vec{c}$,then $x$ is equal to

If $P=(0,1,2)$, $Q=(4,-2,1)$, and $O=(0,0,0)$, then $\angle POQ$ is equal to

If $\vec{a}$ and $\vec{b}$ are two unit vectors with $(\vec{a}, \vec{b}) = \theta$ and $|\vec{a} - \vec{b}| = 1$,then $2|\vec{a} + \vec{b}| \cos \frac{\theta}{2} =$

If $\overrightarrow{a} \cdot \hat{i} = \overrightarrow{a} \cdot (2 \hat{i} + \hat{j}) = \overrightarrow{a} \cdot (\hat{i} + \hat{j} + 3 \hat{k}) = 1$,then $\overrightarrow{a}$ is equal to :

If two vectors $\vec{a}$ and $\vec{b}$ which are perpendicular to each other are such that $|\vec{a}|=8$ and $|\vec{b}|=3$, then $|\vec{a}-2\vec{b}|=$

Vedclass Test Series

Exam Paper Generator

Online Exam Module