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

A set of vectors taken in a given order gives a closed polygon. Then the resultant of these vectors is a 

(1)  scalar quantity

(2)  pseudovector

(3)  unit vector

(4)  null vector

The vector sum of two vectors P and Q is minimum when the angle $\mathrm{\theta }$ between their positive directions, is

(A)  $\frac{\mathrm{\pi }}{4}$

(B)  $\frac{\mathrm{\pi }}{3}$

(C)  $\frac{\mathrm{\pi }}{2}$

(D)  $\mathrm{\pi }$

If the vector sum of two vectors $\overrightarrow{\mathrm{A}}$ and $\overrightarrow{\mathrm{B}}$ is maximum, then the angle $\mathrm{\theta }$ between two vectors will be:

1.  $0°$

2.  $30°$

3.  $45°$

4.  $60°$

If $\overrightarrow{\mathrm{P}}+\overrightarrow{\mathrm{Q}}=\overrightarrow{\mathrm{P}}-\overrightarrow{\mathrm{Q}}$ and $\mathrm{\theta }$ is the angle between $\overrightarrow{\mathrm{P}}$ and $\overrightarrow{\mathrm{Q}}$, then

1.  $\mathrm{\theta }=0°$

2.  $\mathrm{\theta }=90°$

3.  $\mathrm{P}=0$

4.  $\mathrm{Q}=0$

What is the torque of a force $\overrightarrow{\mathrm{F}}=\left(2\hat{\mathrm{i}}-3\hat{\mathrm{j}}+4\hat{\mathrm{k}} \right)$newton acting at a point $\overrightarrow{\mathrm{r}}=\left(3\hat{\mathrm{i}}+2\hat{\mathrm{j}}+3\hat{\mathrm{k}} \right)$ metre about the origin? (Given: $\overrightarrow{\mathrm{\tau }} =\overrightarrow{\mathrm{r} }\times \overrightarrow{\mathrm{F}}$)

1.  $6\hat{\mathrm{i}}-6\hat{\mathrm{j}}+12\hat{\mathrm{k}}$

2.  $17\hat{\mathrm{i}}-6\hat{\mathrm{j}}-13\hat{\mathrm{k}}$

3.  $-6\hat{\mathrm{i}}+6\hat{\mathrm{j}}-12\hat{\mathrm{k}}$

4.  $-17\hat{\mathrm{i}}+6\hat{\mathrm{j}}-13\hat{\mathrm{k}}$

Three non zero vectors $\overrightarrow{\mathrm{A}}, \overrightarrow{\mathrm{B}} \& \overrightarrow{\mathrm{C}}$ satisfy the relation $\overrightarrow{\mathrm{A}}·\overrightarrow{\mathrm{B}}=0 \& \overrightarrow{\mathrm{A}}·\overrightarrow{\mathrm{C}}=0$. Then $\overrightarrow{\mathrm{A}}$ can be parallel to:

(1)  $\overrightarrow{\mathrm{B}}$

(2)  $\overrightarrow{\mathrm{C}}$

(3)  $\overrightarrow{\mathrm{B}}·\overrightarrow{\mathrm{C}}$

(4)  $\overrightarrow{\mathrm{B}}\times \overrightarrow{\mathrm{C}}$

The scalar product of two vectors is 8 and the magnitude of vector product is $8\sqrt{3}$. The angle between them is:

(1)  $30°$

(2)  $60°$

(3)  $120°$

(4)  $150°$