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Two constant forces ${\mathrm{F}}_1=2\hat{\mathrm{i}}-3\hat{\mathrm{j}}+3\hat{\mathrm{k}}<mspace\; width="1em"></mspace>\left(\mathrm{N}\right)$ and ${\mathrm{F}}_2=\hat{\mathrm{i}}+\hat{\mathrm{j}}<mspace\; width="1em"></mspace>+<mspace\; width="1em"></mspace>\hat{\mathrm{k}}$ on a body and displace it from the position ${\mathrm{r}}_1=\hat{\mathrm{i}}+2\hat{\mathrm{j}}-2\hat{\mathrm{k}}<mspace\; width="1em"></mspace>\left(\mathrm{m}\right)$ to the position ${\mathrm{r}}_2=7\hat{\mathrm{i}}+10\hat{\mathrm{j}}+5\hat{\mathrm{k}}<mspace\; width="1em"></mspace>\left(\mathrm{m}\right)$. What is the work done?
1.  30 Joule
2.  41 Joule
3.  -3 Joule
4.  None of these

Two vectors $\mathrm{P}=2\hat{\mathrm{i}}+\mathrm{b}\hat{\mathrm{j}}+2\hat{\mathrm{k}}<mspace\; width="1em"></mspace>\mathrm{and}<mspace\; width="1em"></mspace>\mathrm{Q}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}<mspace\; width="1em"></mspace>\mathrm{will}$ be perpendicular if-
1.  b=0
2.  b=1
3.  b=2
4.  b=-4

Angle that the vector $\mathrm{A}=2\hat{\mathrm{i}}+3\hat{\mathrm{j}}$ makes with y-axis is-
1.  ${\tan }^{-1}<mspace\; width="1em"></mspace>3/2$
2.  ${\tan }^{-1}<mspace\; width="1em"></mspace>2/3$
3.  ${\sin }^{-1}<mspace\; width="1em"></mspace>2/3$
4.  ${\cos }^{-1}<mspace\; width="1em"></mspace>3/2$

The linear velocity of a rotating body is given by v=w$\times$r, where w is the angular velocity and r is the radius vector. The angular velocity of a body $\mathrm{w}=\hat{\mathrm{i}}-2\hat{\mathrm{j}}+2\hat{\mathrm{k}}$ and their radius vector $\mathrm{r}=4\hat{\mathrm{j}}-3\hat{\mathrm{k}},<mspace\; width="1em"></mspace>\left|\mathrm{v}\right|$ is-
1.  $\sqrt{29}$ units
2.  31 units
3.  $\sqrt{37}$ units
4.  $\sqrt{41}$ units

Given the vectors \(\vec{A}=2\hat i+3\hat j-\hat k,\) \(\vec{B}=3\hat i-2\hat j-2\hat k\) and \(\vec C=p\hat i+p\hat j+2p\hat k.\) The angle between \(\vec{A}-\vec{B}\) and \(\vec{C}\) is:
1. \(\theta=\cos^{-1}\left(\dfrac{2}{\sqrt3} \right) \)
2. \(\theta=\cos^{-1}\left(\dfrac{\sqrt3}{2} \right) \)
3. \(\theta=\cos^{-1}\left(\dfrac{\sqrt2}{3} \right) \)
4. none of these

Component of $3\hat{\mathrm{i}}+4\hat{\mathrm{j}}$ perpendicular to $\hat{\mathrm{i}}+\hat{\mathrm{j}}$ and in the same plane as that of $3\hat{\mathrm{i}}+4\hat{\mathrm{j}}$ is:
1. $\frac{1}{2}\left(\hat{\mathrm{j}}-\hat{\mathrm{i}}\right)$
2. $\frac{3}{2}\left(\hat{\mathrm{j}}-\hat{\mathrm{i}}\right)$
3. $\frac{5}{2}\left(\hat{\mathrm{j}}-\hat{\mathrm{i}}\right)$
4. $\frac{7}{2}\left(\hat{\mathrm{j}}-\hat{\mathrm{i}}\right)$