Angle between Two Vectors

Q. Two adjacent sides of a parallelogram ABCD are given by $\stackrel{⟶}{AB}=2\hat{i}+10\hat{j}+11\hat{k}$ and $\stackrel{⟶}{AD}=-\hat{i}+2\hat{j}+2\hat{k}.$
The side AD is rotated by an acute angle $\alpha$ in the plane of the parallelogram so that AD becomes AD'. If AD' makes a right angle with the side AB, then the cosine of the angle $\alpha$ is given by

1. $\frac{8}{9}$
2. $\frac{\sqrt{17}}{9}$
3. $\frac{1}{9}$
4. $\frac{4\sqrt{5}}{9}$

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. 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 $\overrightarrow{a}$ 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 $\overrightarrow{a}$ and $\overrightarrow{b}$ are two unit vectors such that $\overrightarrow{a}+2\overrightarrow{b}$ and $5\overrightarrow{a}-4\overrightarrow{b},$ are perpendicular to each other, then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is

1. ${60}^{\circ }$
2. $co{s}^{-1}\left(\frac{2}{7}\right)$
3. ${45}^{\circ }$
4. $co{s}^{-1}\left(\frac{1}{3}\right)$

Q.

$\overrightarrow{a}$ and $\overrightarrow{c}$ are unit vectors and $|\overrightarrow{b}|=4.$ The angle between $\overrightarrow{a}and\overrightarrow{c}isco{s}^{-1}\left(\frac{1}{4}\right).$
If $\overrightarrow{b}-2\overrightarrow{c}=\lambda \overrightarrow{a}$, then $\lambda$ is

1. $3or-4$

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

3. $-3or4$

4. $-\frac{1}{4}or\frac{3}{4}$

Q. If $a=-\hat{i}+\hat{j}+2\hat{k}$, $b=2\hat{i}-\hat{j}-\hat{k}$ then $c=-2\hat{i}+\hat{j}+3\hat{k}$, then the angle between 2a-c and a+b is:

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

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. The area of the parallelogram formed by the tangents at the points whose eccentric angles are $\theta ,\theta +\frac{\pi }{2},\theta +\pi ,\theta +\frac{3\pi }{2}$ on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is

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

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

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

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