Unit Vectors

Q. Let $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ be three unit vectors such that ${|\overrightarrow{a}-\overrightarrow{b}|}^2+{|\overrightarrow{a}-\overrightarrow{c}|}^2=8.$Then ${|\overrightarrow{a}+2\overrightarrow{b}|}^2+{|\overrightarrow{a}+2\overrightarrow{c}|}^2$ is equal to:

Q. If a, b and c are unit vectors, then $|a-b{|}^2+|b-c{|}^2+|c-a{|}^2$ does not exceed

1. 4
2. 9
3. 8
4. 6

Q. Which of the following is a unit vector in the direction of $\hat{i}+\hat{j}+\hat{k}$.

1. 
2. 
3. 
4. None of these

Q. Which of the following is a unit vector

1. 
2. (1, 1)
3. 
4. 

Q. Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be two non-collinear unit vectors. If $\overrightarrow{u}=\overrightarrow{a}-(\overrightarrow{a}.\overrightarrow{b})\overrightarrow{b}$ and $\overrightarrow{v}=\overrightarrow{a}\times \overrightarrow{b},$ then $|\overrightarrow{v}|$ is

1. $|\overrightarrow{u}|$
2. $|\overrightarrow{u}|+|\overrightarrow{u}.\overrightarrow{a}|$
3. $|\overrightarrow{u}|+|\overrightarrow{u}.\overrightarrow{b}|$
4. $|\overrightarrow{u}|+\overrightarrow{u}.(\overrightarrow{a}+\overrightarrow{b})$

Q.

Given $\overrightarrow{A}=0.3\hat{i}+0.4\hat{j}+c\hat{k}$. Calculate the value of c if A is a unit vector.

1. 0.45

2. 0.75

3. 0.87

4. 0.96

Q. If a unit vector $\overrightarrow{a}$ makes an angle $\frac{\pi }{3}$ with $\hat{i}$, $\frac{\pi }{4}$ with $\hat{j}$ and $\theta \in (0,\pi )$ with $\hat{k}$, then a value of $\theta$ is :

1. $\frac{5\pi }{12}$
2. $\frac{\pi }{4}$
3. $\frac{2\pi }{3}$
4. $\frac{5\pi }{12}$

Q. A unit vector $\overrightarrow{a}$ makes an angle of $45°$ with positive $z-$axis and if $\overrightarrow{a}+\hat{i}+\hat{j}$ is a unit vector, then $\overrightarrow{a}=$

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

Q. If the unit vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are inclined at an angle 2 $\theta$ such that $|\overrightarrow{a}-\overrightarrow{b}|$ <1 and 0 $\le \theta \le \pi$, then $\theta$ lies in the interval

1. $[0,\frac{\pi }{6})$
2. $(\frac{5\pi }{6},\pi ]$
3. $[\frac{\pi }{6},\frac{\pi }{2}]$
4. $[\frac{\pi }{2},\frac{\pi }{6}]$

Q. If the unit vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are inclined at an angle 2 $\theta$ such that $|\overrightarrow{a}-\overrightarrow{b}|$ <1 and 0 $\le \theta \le \pi$, then $\theta$ lies in the interval

1. $[0,\frac{\pi }{6})$
2. $(\frac{5\pi }{6},\pi ]$
3. $[\frac{\pi }{6},\frac{\pi }{2}]$
4. $[\frac{\pi }{2},\frac{\pi }{6}]$