Component of $3\hat{i}+4\hat{j}$ perpendicular to $\hat{i}+\hat{j}$ and in the same plane as that of $3\hat{i}+4\hat{j}$ is:

1. $\frac{1}{2}(\hat{j}-\hat{i})$

2. $\frac{3}{2}(\hat{j}-\hat{i})$

3. $\frac{5}{2}(\hat{j}-\hat{i})$

4. $\frac{7}{2}(\hat{j}-\hat{i})$

If vectors A = cosωt $\hat{i}$ + sinωt $\hat{j}$ and B = (cosωt/2) $\hat{i}$ + (sinωt/2) $\hat{j}$ are functions of time, then the value of t at which they are orthogonal to each other

1. t=$\mathrm{\pi }$/4ω
2. t=$\mathrm{\pi }$/2ω
3. t=$\mathrm{\pi }$/ω
4. t=0 

 

A particle moves from a point \(\left(\right. - 2 \hat{i} + 5 \hat{j} \left.\right)\) to \(\left(\right. 4 \hat{j} + 3 \hat{k} \left.\right)\) when a force of \(\left(\right. 4 \hat{i} + 3 \hat{j} \left.\right)\) \(\text{N}\) is applied. How much work has been done by the force?

Two constant forces ${\overrightarrow{\mathrm{F}}}_1=2\hat{\mathrm{i}}-3\hat{\mathrm{j}}+3\hat{\mathrm{k}} \left(\mathrm{N}\right)$ and ${\overrightarrow{\mathrm{F}}}_2=\hat{\mathrm{i}}+\hat{\mathrm{j}}-2\hat{\mathrm{k}} \left(\mathrm{N}\right)$ act on a body and displace it from the position ${\overrightarrow{\mathrm{r}}}_1=\hat{\mathrm{i}}+2\hat{\mathrm{j}}-2\hat{\mathrm{k}} \left(\mathrm{m}\right)$ to the position ${\overrightarrow{\mathrm{r}}}_2=7\hat{\mathrm{i}}+10\hat{\mathrm{j}}+5\hat{\mathrm{k}} \left(\mathrm{m}\right)$. What is the work done W? $[Given : W =\overrightarrow{F}.∆\overrightarrow{r}]$

(A)  9 Joule

(B)  41 Joule

(C)  -3 Joule

(D)  None of these

Given the vectors

$\overrightarrow{\mathrm{A}}=2\hat{\mathrm{i}}+3\hat{\mathrm{j}}-\hat{\mathrm{k}}$
$\overrightarrow{\mathrm{B}}=3\hat{\mathrm{i}}-2\hat{\mathrm{j}}-2\hat{\mathrm{k}}$
$\& \overrightarrow{\mathrm{C}}=\mathrm{p}\hat{\mathrm{i}}+\mathrm{p}\hat{\mathrm{j}}+2\mathrm{p}\hat{\mathrm{k}}$

Find the angle between $\left(\overrightarrow{\mathrm{A}}-\overrightarrow{\mathrm{B}}\right) \& \overrightarrow{\mathrm{C}}$

(A)  $\mathrm{\theta }={\cos }^{-1}\left(\frac{2}{\sqrt{3}}\right)$

(B)  $\mathrm{\theta }={\cos }^{-1}\left(\frac{\sqrt{3}}{2}\right)$

(C)  $\mathrm{\theta }={\cos }^{-1}\left(\frac{\sqrt{2}}{3}\right)$

(D)  none of these

The vector having a magnitude of 10 and perpendicular to the vector $3\hat{\mathrm{i}}-4\hat{\mathrm{j}}$ is- 

1. $4\hat{\mathrm{i}}-3\hat{\mathrm{j}}$

2. $5\sqrt{2} \hat{\mathrm{i}}-5\sqrt{2} \hat{\mathrm{j}}$

3. $8\hat{\mathrm{i}}+6\hat{\mathrm{j}}$

4.  $8\hat{\mathrm{i}}-6\hat{\mathrm{j}}$

A force $\overrightarrow{\mathrm{F}}=3\hat{\mathrm{i}}+\mathrm{c}\hat{\mathrm{j}}+2\hat{\mathrm{k}}$ acting on a particle causes a displacement $\overrightarrow{\mathrm{d}}=-4\hat{\mathrm{i}}+2\hat{\mathrm{j}}+3\hat{\mathrm{k}}$. If the work done is 6J then the value of 'c' is- $[Given: W=\overrightarrow{F.}\overrightarrow{d}]$

1. 12 

2. 0

3. 6

4. 1

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

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