Angle between Two Vectors

Q. If the angle θ between the vectors $a=2x^2\hat{i}+4x\hat{j}+\hat{k}$ and $b=7\hat{i}-2\hat{j}+x\hat{k}$ is such that ${90}^{\circ }$ < $\theta$ < ${180}^{\circ }$
then x lies in the interval:

1. $(\frac{1}{2},1)$
2. $(0,\frac{1}{2})$
3. $(1,\frac{3}{2})$
4. $(\frac{1}{2},\frac{3}{2})$

Q. Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be two vectors such that $|\overrightarrow{a}|=1,$ $|\overrightarrow{b}|=4$ and $\overrightarrow{a}\cdot \overrightarrow{b}=2.$ If $\overrightarrow{c}=(2\overrightarrow{a}\times \overrightarrow{b})-3\overrightarrow{b},$ then the angle between $\overrightarrow{b}$ and $\overrightarrow{c}$ is

1. $\frac{5\pi }{6}$
2. $\frac{2\pi }{3}$
3. $\frac{\pi }{6}$
4. $\frac{\pi }{3}$

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

Q. If the vectors $\overrightarrow{a}=\left(clo{g}_{2}x\right)\hat{i}-6\hat{j}+2\hat{k}$ and $\overrightarrow{b}=\left(lo{g}_{2}x\right)\hat{i}+2\hat{j}+3\left(clo{g}_{2}x\right)\hat{k}$ make an obtuse angle for any $x\in (0,\infty )$ then c belongs to

1. $(-\infty ,-\frac{4}{3})$
2. $(-\infty ,0)$
3. $(-\frac{4}{3},0)$
4. $(-\frac{4}{3},\infty )$

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

Q. If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are three mutually perpendicular vectors of equal magnitude, then the angle $\theta$ which $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}$ makes with any one of three given vectors is given by

1. $co{s}^{-1}\left(\frac{1}{\sqrt{3}}\right)$
2. $co{s}^{-1}\left(\frac{1}{3}\right)$
3. $co{s}^{-1}\left(\frac{2}{\sqrt{3}}\right)$
4. $Noneofthese$

Q. If $\overrightarrow{a},\overrightarrow{b}$ are unit vectors such that the vector $\overrightarrow{a}+3\overrightarrow{b}$ is perpendicular to $7\overrightarrow{a}-5\overrightarrow{b}$ and $\overrightarrow{a}-4\overrightarrow{b}$ is perpendicular to $7\overrightarrow{a}-2\overrightarrow{b}$, then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is

1. $\frac{\pi }{6}$
2. $\frac{\pi }{4}$
3. $\frac{\pi }{3}$
4. $\frac{\pi }{2}$

Q. If $\overrightarrow{a}$ and $\overrightarrow{b}$ are two vectors such that $\overrightarrow{a}.\overrightarrow{b}=0$ and $\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{0}$, then

1. $Noneoftheabove$
2. $eitherparallelorperpedicular$
3. $\overrightarrow{a}||\overrightarrow{b}$
4. $\overrightarrow{a}\perp \overrightarrow{b}$

Q. The points O, A, B, C, D are such that $\overline{OA}=a,\overline{OB}=b,\overline{OC}=2a+3b$ and $\overline{OD}=a-2b$. If $|a|=3|b|$, then the angle between $\overline{BD}$ , $\overline{AC}$ is

1. $\frac{\pi }{3}$
2. $\frac{\pi }{4}$
3. $\frac{\pi }{6}$
4. $noneofthese$

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

1. 
2. 
3. 
4.