Multiplication with Vectors

Q. Which of the following is the unit vector perpendicular to $\overrightarrow{A}$ and $\overrightarrow{B}$?

1. $\frac{\hat{A}\times \hat{B}}{AB\sin \theta }$
2. $\frac{\hat{A}\times \hat{B}}{AB\cos \theta }$
3. $\frac{\overrightarrow{A}\times \overrightarrow{B}}{AB\sin \theta }$
4. $\frac{\overrightarrow{A}\times \overrightarrow{B}}{AB\cos \theta }$
   

Q. Projection of $\overrightarrow{A}$ along the vector $\overrightarrow{B}$ is

1. $\frac{\overrightarrow{A}\cdot \overrightarrow{B}}{|\overrightarrow{B}|}$
2. $\frac{\overrightarrow{A}\times \overrightarrow{B}}{|\overrightarrow{B}|}$
3. $\frac{\overrightarrow{A}\cdot \overrightarrow{B}}{|\overrightarrow{A}|}$
4. $\frac{\overrightarrow{A}\cdot \overrightarrow{B}}{\overrightarrow{B}}$

Q.

If for two vector $\overrightarrow{\mathrm{A}}$ and $\overrightarrow{\mathrm{B}}$, sum $\left(\overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}}\right)$ is perpendicular to the difference $\left(\overrightarrow{\mathrm{A}}-\overrightarrow{\mathrm{B}}\right)$. The ratio of their magnitude is

1. $1$

2. $2$

3. $3$

4. None of these

Q. A vector $\overrightarrow{F_1}$ is along $x$-axis. If its vector product with another vector $\overrightarrow{F_2}$ is zero, then $\overrightarrow{F_2}$ could be

1. $4\hat{j}$
2. $-(\hat{i}+\hat{j})$
3. $\hat{j}+\hat{k}$
4. $-4\hat{i}$

Q. If vector$\left(\hat{i}+\hat{j}+\hat{k}\right)$ and $3\hat{i}$ form two sides of a triangle and the magnitude of area of triangle is $\frac{x}{\sqrt{y}}$, then x+y is
(x and y are prime numbers).

Q. The area of the parallelogram represented by the vectors $\overrightarrow{A}=2\hat{i}+3\hat{j}$ and $\overrightarrow{B}=\hat{i}+4\hat{j}$ as its sides is

1. $15\text{Units}$
2. $7.5\text{Units}$
3. $10\text{Units}$
4. $5\text{Units}$

Q.

What is the angle between the following pair vectors? $\overrightarrow{A}=\hat{i}+\hat{j}+\hat{k}and\overrightarrow{B}=-2\hat{i}-2\hat{j}-2\hat{k}$

1. ${30}^{\circ }$

2. ${60}^{\circ }$

3. ${90}^{\circ }$

4. ${180}^{\circ }$

Q. Two forces $\overrightarrow{F_1}=(2\hat{i}+2\hat{j})N$ and $\overrightarrow{F_2}=(3\hat{j}+4\hat{k})N$ are acting on a particle. The angle between $\overrightarrow{F_1}\text{and}\overrightarrow{F_2}$ is

1. $\theta =co{s}^{-1}\left(\frac{3}{2\sqrt{5}}\right)$
2. $\theta =co{s}^{-1}\left(\frac{3}{5\sqrt{2}}\right)$
3. $\theta =co{s}^{-1}\left(\frac{2}{3\sqrt{5}}\right)$
4. $\theta =co{s}^{-1}\left(\frac{\sqrt{3}}{5}\right)$

Q.

Find the area of the shaded region.
$r=\sqrt{\theta }$

Q.

Why cross product is a vector quantity ?

Q. The vectors from the origin $O$ to the points $A$ and $B$ is$\overrightarrow{A}=3\hat{i}-6\hat{j}+2\hat{k}$ and $\overrightarrow{B}=2\hat{i}+\hat{j}-2\hat{k}$ respectively. The area of triangle $OAB$ be

1. $\frac{5}{2}\sqrt{17}$ sq. units
2. $\frac{2}{5}\sqrt{17}$ sq. units
3. $\frac{3}{5}\sqrt{17}$ sq. units
4. $\frac{5}{3}\sqrt{17}$ sq. units

Q. Two vectors are mutually perpendicular if their

1. dot product is zero.
2. cross product is zero.
3. resultant is zero.
4. none of these

Q.

How do you find the cross product of vectors $?$

Q. Two vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ are at right angles to each other , when

1. $\overrightarrow{A}+\overrightarrow{B}=0$
2. $\overrightarrow{A}-\overrightarrow{B}=0$
3. $\overrightarrow{A}\times \overrightarrow{B}=0$
4. $\overrightarrow{A}.\overrightarrow{B}=0$

Q. The angle between vectors $\overrightarrow{A}\times \overrightarrow{B}$ and $\overrightarrow{B}\times \overrightarrow{A}$ is

1. $0$
2. ${180}^{\circ }$
3. ${60}^{\circ }$
4. ${30}^{\circ }$

Q.

If $a$ and $b$ are two non-zero, then a vector perpendicular to the vector $(b.b)a-(a.b)b$ is?

1. $a$

2. $b$

3. $a–b$

4. $a+b$

Q. If $\overrightarrow{A}\times \overrightarrow{B}=\overrightarrow{B}\times \overrightarrow{A}$, then the angle between $\overrightarrow{A}$ and $\overrightarrow{B}$ is

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

Q.

What is the dot product of a vector with itself?

Q. If $\overrightarrow{A}\times \overrightarrow{B}=\overrightarrow{C}$, then choose the incorrect option.

1. $\overrightarrow{C}\perp (\overrightarrow{A}+\overrightarrow{B})$
2. $\overrightarrow{C}\perp (\overrightarrow{A}-\overrightarrow{B})$
3. $\overrightarrow{C}\perp (\overrightarrow{A}\times \overrightarrow{B})$
4. $\overrightarrow{C}\perp \overrightarrow{A}$ and $\overrightarrow{B}$

Q. If the scalar product of two vectors is negative, then the angle between them has the range

1. $0\le \theta <{90}^{\circ }$
2. $\theta ={90}^{\circ }$
3. ${90}^{\circ }<\theta \le {180}^{\circ }$
4. None of these

Q.

How do you multiply vectors by vectors$?$

Q.

If $\overrightarrow{A}\times \overrightarrow{B}=\overrightarrow{C}$, then which of the following is wrong

1. $\overrightarrow{C}\perp \overrightarrow{A}$

2. $\overrightarrow{C}\perp \overrightarrow{B}$

3. $\overrightarrow{C}\perp \left(\overrightarrow{A}+\overrightarrow{B}\right)$

4. $\overrightarrow{C}\perp \left(\overrightarrow{A}\times \overrightarrow{B}\right)$

Q. Find $\overrightarrow{A}\times \overrightarrow{B}$, if $\overrightarrow{A}=3\hat{i}-\hat{j}+3\hat{k}$ and $\overrightarrow{B}=-2\hat{i}+3\hat{j}+4\hat{k}$

1. $13\hat{i}-18\hat{j}+7\hat{k}$
2. $-13\hat{i}-18\hat{j}-7\hat{k}$
3. $-13\hat{i}+18\hat{j}+7\hat{k}$
4. $-13\hat{i}-18\hat{j}+7\hat{k}$

Q. The position vector of a particle is given as $\overrightarrow{r}=(t^2-4t+6)\hat{i}+\left(t^2\right)\hat{j}.$ The time after which the velocity vector and acceleration vector becomes perpendicular to each other is equal to

1. $1\text{sec}$
2. $1.5\text{sec}$
3. $2\text{sec}$
4. $\text{Not Possible}$

Q. Find $\overrightarrow{A}\times \overrightarrow{B}$, if $\overrightarrow{A}=2\hat{i}-\hat{j}+3\hat{k}$ and $\overrightarrow{B}=-2\hat{i}+3\hat{j}+4\hat{k}$

1. $-13\hat{i}-14\hat{j}-4\hat{k}$
2. $5\hat{i}+14\hat{j}+4\hat{k}$
3. $5\hat{i}-14\hat{j}+4\hat{k}$
4. $-13\hat{i}-14\hat{j}+4\hat{k}$

Q. Projection of a vector onto itself is

1. equal to $0$.
2. equal to $1$.
3. equal to $-1$.
4. equal to its magnitude.

Q. Two vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ have magnitude $3$ each. If $\overrightarrow{A}\times \overrightarrow{B}=-5\hat{k}+2\hat{i}$, then the angle between $A$ and $B$ is

1. ${\sin }^{-1}\left(\frac{2}{5}\right)$
2. ${\sin }^{-1}\left(\frac{\sqrt{29}}{9}\right)$
3. ${\tan }^{-1}\left(\frac{-5}{2}\right)$
4. ${\cos }^{-1}\left(\frac{\sqrt{29}}{9}\right)$

Q. The rectangular components of a vector are $(2,2)$. The corresponding rectangular components of another vector are $(1,\sqrt{3})$ . Find the angle between the two vectors (in degree).

Q. A force $(3\hat{i}+4\hat{j})N$ acts on a body and displaces it by $(3\hat{i}+4\hat{j})$ meters. The work done by the force is
(Given that formula for work done is $W=\overrightarrow{F}.\overrightarrow{s})$

1. $5J$
2. $25J$
3. $10J$
4. $30J$

Q.

If $\cos ec\theta =cothx$, then the value of $\tan \theta$ is

1. $\cos hx$

2. $\sin hx$

3. $\tan hx$

4. $\cos echx$