Angle between Two Vectors

Q.

What is a $3$d angle?

Q.

A line $AB$ in three-dimensional space makes angles $45°and120°$ with the positive x-axis and the positive y-axis respectively. If $AB$ makes an acute angle $\theta$ with the positive respectively. If $AB$ makes an acute angle $\theta$ with the positive z-axis, then $\theta$ equals

1. $30°$

2. $45°$

3. $60°$

4. $75°$

Q. What is the angle between the two vectors shown in the figure?

1. $\frac{\theta }{2}$
2. $\theta$
3. $\pi -\theta$
4. $\theta -\pi$

Q. The vectors $\overrightarrow{a}=3\hat{i}-2\hat{j}+2\hat{k}$ and $\overrightarrow{b}=-\hat{i}-2\hat{k}$ are the adjacent sides of a parallelogram.Then, angle between its diagonals is

1. $\frac{\pi }{4}$
2. $\frac{2\pi }{3}$
3. $\frac{\pi }{3}$
4. $\frac{\pi }{2}$

Q. If $\theta$ and $\varphi$ are positive angles such that $\theta +\varphi =\frac{\pi }{4}$, then maximum value of $\tan \theta \tan \varphi$ is

1. $\sqrt{2}+1$
2. $\frac{\sqrt{2}+1}{\sqrt{2}-1}$
3. $\frac{\sqrt{2}-1}{\sqrt{2}+1}$
4. $\sqrt{2}-1$

Q. The value of $A_i{A}_j$ where $j=1,2,3,...n$ is

1. $2R\sin \left[(j-1)\frac{\pi }{2n}\right]$
2. $R\sin \left[(j-1)\frac{2\pi }{n}\right]$
3. $2R\sin \left[(j-1)\frac{\pi }{n}\right]$
4. $R\sin \left[(j-1)\frac{\pi }{n}\right]$

Q. If $\overrightarrow{a}$ and $\overrightarrow{b}$ are two unit vectors inclined at an angle $\theta$ such that $\overrightarrow{a}+\overrightarrow{b}$ is a unit vector, then $\theta$ is equal to

1. $\frac{\pi }{3}$
2. $\frac{\pi }{4}$
3. $\frac{\pi }{2}$
4. $\frac{2\pi }{3}$