The x and y components of vector $\overrightarrow{\mathrm{A}}$ are 4m and 6m respectively. The x, y components of vector $\overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}}$ are 10m and 9m respectively. The length of $\overrightarrow{\mathrm{B}}$ is ______ and angle that $\overrightarrow{\mathrm{B}}$ makes with the x axis is given by _______.

(A)  $\sqrt{0.45}\mathrm{m}, {\tan }^{-1}\left(1/2\right)$

(B)  $\sqrt{4.5}\mathrm{m}, {\tan }^{-1}\left(1/2\right)$

(C)  $\sqrt{45}\mathrm{m}, {\tan }^{-1}\left(1/2\right)$

(D)  $4\sqrt{5}\mathrm{m}, {\tan }^{-1}\left(1/2\right)$

A particle travels with speed 50 m/s from the point (3, 7) in a direction $7\hat{\mathrm{i}}-24\hat{\mathrm{j}}$. Find its position vector after 3 seconds.

1.  $151\hat{\mathrm{i}}+45\hat{\mathrm{j}}$

2.  $45\hat{\mathrm{i}}-137\hat{\mathrm{j}}$

3.  $151\hat{\mathrm{i}}-45\hat{\mathrm{j}}$

4.  $4.5\hat{\mathrm{i}}-151\hat{\mathrm{j}}$

If \(\overrightarrow{\mathbf{A}}{=}{2}\hat{i}+\hat{j}\;{\&}\;\overrightarrow{\mathbf{B}}{=}\hat{i}{-}\hat{j}\) , then the components of $\overrightarrow{\mathrm{A}}$ along with $\overrightarrow{\mathrm{B}}$ & perpendicular to $\overrightarrow{\mathrm{B}}$ respectively will be:
1.  $\frac{\hat{\mathrm{i}}-\hat{\mathrm{j}}}{2}, \frac{3}{2}\left(\hat{\mathrm{i}}+\hat{\mathrm{j}}\right)$

2.  $\frac{\hat{\mathrm{i}}-\hat{\mathrm{j}}}{2},- \frac{2}{3}\left(\hat{\mathrm{i}}+\hat{\mathrm{j}}\right)$

3.  $\frac{\hat{\mathrm{i}}-\hat{\mathrm{j}}}{2}, -\frac{3}{2}\left(\hat{\mathrm{i}}-\hat{\mathrm{j}}\right)$

4.  $\frac{\hat{\mathrm{i}}-\hat{\mathrm{j}}}{2}, \frac{2}{3}\left(\hat{\mathrm{i}}-\hat{\mathrm{j}}\right)$

If \(\vec{a}\) is a vector and \(x\) is a non-zero scalar, then which of the following is correct?
1. \(x\vec{a}\) is a vector in the direction of \(\vec{a}\)
2. \(x\vec{a}\) is a vector collinear to \(\vec{a}\)
3. \(x\vec{a}\) and \(\vec{a}\) have independent directions
4. \(x\vec{a}\) is a vector perpendicular to \(\vec{a}\)

If \(\theta\) is the angle between two vectors $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$, and $|\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}}|=\overrightarrow{\mathrm{a}}.\overrightarrow{\mathrm{b}}$, then \(\theta\) is equal to:
1. \(0^\circ\)
2. \(180^\circ\)
3. \(135^\circ\)
4. \(45^\circ\)

The vector $\overrightarrow{\mathrm{b}}$, which is collinear with the vector $\overrightarrow{\mathrm{a}}$=(2, 1, -1) and satisfies the condition $\overrightarrow{\mathrm{a}}$.$\overrightarrow{\mathrm{b}}$=3 is-

1.  (1, 1/2, -1/2)

2.  (2/3, 1/3, -1/3)

3.  (1/2, 1/4, -1/4)

4.  (1, 1, 0)

If a, b and c are three non-zero vectors such that $\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}=0$, then the value of $\overrightarrow{\mathrm{a}}.\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{b}}.\overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{c}}.\overrightarrow{\mathrm{a}}$ will be:

1.  Less than zero

2.  equal to zero

3.  greater than zero

4.  3

Two vectors $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$ inclined at an angle $\mathrm{\theta }$ with respect to each other have a resultant $\overrightarrow{\mathrm{c}}$ which makes an angle $\mathrm{\beta }$ with $\overrightarrow{\mathrm{a}}$. If the directions of $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$ are interchanged, then the resultant will have the same

(A)  magnitude

(B)  direction

(C)  magnitude as well as direction

(D)  neither magnitude nor direction

Two vectors $\overrightarrow{\mathrm{A}}$ and $\overrightarrow{\mathrm{B}}$ lie in a plane. Another vector $\overrightarrow{\mathrm{C}}$ lies outside this plane. The resultant $\overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}}+\overrightarrow{\mathrm{C}}$ of these three vectors

(1)  can be zero

(2)  cannot be zero

(3)  lies in the plane of $\overrightarrow{\mathrm{A}} \&\overrightarrow{\mathrm{B}}$

(4)  lies in the plane of $\overrightarrow{\mathrm{A}} \& \overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}}$

The vector sum of the forces of 10 N and 6 N can be

(1)  2 N

(2)  8 N

(3)  18 N

(4)  20 N