Dot Product of Two Vectors

Q. $\mathrm{If}\overrightarrow{A}\mathrm{and}\overrightarrow{B}\mathrm{are}\mathrm{two}\mathrm{unit}\mathrm{vectors}\mathrm{such}\mathrm{that}3\overrightarrow{A}+4\overrightarrow{B}\mathrm{and}5\overrightarrow{A}-3\overrightarrow{B}\mathrm{are}\mathrm{perpendicular},\mathrm{then}\mathrm{the}\mathrm{angle}\mathrm{between}\overrightarrow{A}\mathrm{and}\overrightarrow{B}\mathrm{is}$

1. ${90}^o$
2. ${\cos }^{-1}\left(\frac{-3}{11}\right)$
3. ${\sin }^{-1}\left(\frac{-3}{11}\right)$
4. $0^o$

Q.

If $\overline{a}and\overline{b}$ are two vectors as shown in the figure, then $\overline{a}.\overline{b}$ will equal to $|\overline{a}||\overline{b}|cos\theta$

1. True

2. False

Q.

Suppose that p, q and r are three non-coplanar vectors in $R^3$. Let the components of a vector s along p, q and r be 4, 3 and 5 respectively. If the components of this vector s along (-p+q+r), (p-q+r) and (-p-q+r) are $x$, $y$ and $z$ respectively, then the value of $2x+y+z$ is ___

Q.

$Themagnitudeoftheprojectionof2\hat{i}+3\hat{j}+4\hat{k}onthevector\hat{i}+\hat{j}+\hat{k}willbe\underset{–}{}$

1. $\sqrt{3}$

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

3. $3\sqrt{3}$

4. None of the above

Q. Let $\overrightarrow{a}=\hat{i}+\alpha \hat{j}+3\hat{k}$ and $\overrightarrow{b}=3\hat{i}-\alpha \hat{j}+\hat{k}$. If the area of the parallelogram whose adjacent sides are represented by the vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $8\sqrt{3}$ square units, then $\overrightarrow{a}\cdot \overrightarrow{b}$ is equal to

Q.

The angle between the vectors $\overline{u}=<3,0>and\overline{v}=<5,5>$ is ___

Q. Let $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ be three non-coplanar unit vectors such that the angle between each pair of vectors is $\frac{\pi }{3}.$ If $\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{b}\times \overrightarrow{c}=p\overrightarrow{a}+q\overrightarrow{b}+r\overrightarrow{c},$ where $p,q$ and $r$ are scalars, then the value of $\frac{p^2+2q^2+r^2}{q^2}$ is

Q. Let $A,B$ are two points on the curve $y={\log }_{1/2}(x-\frac{1}{2})+{\log }_2\sqrt{4x^2-4x+1}$ and $A$ is also on the circle whose radius is $\sqrt{10}$ and centre at $O(0,0)$. $B$ lies inside the circle such that its abscissa is integer, then

1. $\overrightarrow{OA}\cdot \overrightarrow{OB}=4$
2. $\overrightarrow{OA}\cdot \overrightarrow{OB}=7$
3. $|\overrightarrow{AB}|=2$
4. $|\overrightarrow{AB}|=1$

Q.

Let two non-collinear unit vectors $\hat{a}\text{and}\hat{b}$ form an acute angle. A point P moves so that at any time t the position vector
$\overrightarrow{OP}$ (where, O is the origin) is given by
$\hat{a}\cos t+\hat{b}\sin t$. When P is farthest from origin O, let M be the length of $\overrightarrow{OP}$ and $\hat{u}$ be the unit vector along $\overrightarrow{OP}$. Then,

1. $\hat{u}=\frac{\hat{a}+\hat{b}}{|\hat{a}+\hat{b}|}\text{and}M=(1+\hat{a}.\hat{b}{)}^{\frac{1}{2}}$
   

2. $\hat{u}=\frac{\hat{a}-\hat{b}}{|\hat{a}-\hat{b}|}\text{and}M=(1+\hat{a}.\hat{b}{)}^{\frac{1}{2}}$

3. $\hat{u}=\frac{\hat{a}+\hat{b}}{|\hat{a}+\hat{b}|}\text{and}M=(1+2\hat{a}.\hat{b}{)}^{\frac{1}{2}}$

4. $\hat{u}=\frac{\hat{a}-\hat{b}}{|\hat{a}-\hat{b}|}\text{and}M=(1+2\hat{a}.\hat{b}{)}^{\frac{1}{2}}$

Q.

If $\overrightarrow{a}\times (\overrightarrow{b}\times \overrightarrow{c})$ is perpendicular to $(\overrightarrow{a}\times \overrightarrow{b})\times \overrightarrow{c}$ then, we may have,

1. $\overrightarrow{b}.\overrightarrow{c}=0$

2. $\overrightarrow{a}.\overrightarrow{b}=0$

3. $\overrightarrow{a}.\overrightarrow{c}=0$

4. None of these

Q.

Suppose that p, q and r are three non-coplanar vectors in $R^3$. Let the components of a vector s along p, q and r be 4, 3 and 5 respectively. If the components of this vector s along (-p+q+r), (p-q+r) and (-p-q+r) are $x$, $y$ and $z$ respectively, then the value of $2x+y+z$ is ___

Q.

$Aforceof\overrightarrow{F}=3\hat{i}+4\hat{j}isactingonaboxatpoint\overrightarrow{A}whosepositionvectorwithrespecttooriginis<2,3>.Workdoneindisplacingtheparticlefrom\overrightarrow{A}to\overrightarrow{B}whosepositionvectorwithrespecttooriginis<5,6>willbe.......units$

__

Q.

George is pulling Cameron on a toboggan and is exerting a force of 40N acting at an angle of ${60}^{\circ }$ to the ground. If Cameron is pulled by a distance of 100 m horizontally, then the work done by George is __ J

Q.

Let P, Q, R and S be the points on the plane with position vectors
$-2\hat{i}-\hat{j},4\hat{i},3\hat{i}+3\hat{j}and-3\hat{i}+2\hat{j}$, respectively. The quadrilateral PQRS must be a

1. parallelogram, which is neither a rhombus nor a rectangle

2. square

3. rectangle, but not a square

4. rhombus, but not a square

Q. If $\hat{a},\hat{b}$ and $\hat{c}$ are unit vectors, and the maximum value of ${|2\hat{a}-3\hat{b}|}^2+{|2\hat{b}-3\hat{c}|}^2+{|2\hat{c}-3\hat{a}|}^2$ is $p,$ then the value of $\left[\frac{p}{10}\right]$ is
(Here, $[.]$ denotes the greatest integer function.)

Q.

If $\overrightarrow{a}=x\hat{i}+(x-1)\hat{j}+\hat{k}$ and $\overrightarrow{b}=(x+1)\hat{i}+\hat{j}+a\hat{k}$ always make an acute angle with each other for every value of $xϵR$, then

1. $aϵ(-\infty ,2)$

2. $aϵ(2,\infty )$

3. $aϵ(-\infty ,1)$

4. $aϵ(1,\infty )$

Q.

$If\overline{a}.\overline{b}=\overline{a}.\overline{c}and\overline{a}\ne \overline{0},thenwecansaythat\overline{b}=\overline{c}.$

1. True

2. False

Q.

If $\overrightarrow{a}\times (\overrightarrow{b}\times \overrightarrow{c})$ is perpendicular to $(\overrightarrow{a}\times \overrightarrow{b})\times \overrightarrow{c}$ then, we may have,

1. $\overrightarrow{b}.\overrightarrow{c}=0$

2. $\overrightarrow{a}.\overrightarrow{b}=0$

3. $\overrightarrow{a}.\overrightarrow{c}=0$

4. None of these