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

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(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 $3i - 4j$.

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