If the vectors PQ = - 3i + 4j + 4k and PR = 5i - 2j + 4k are the sides of a $∆$PQR, then the length of the median through P is

• $\sqrt{14}$

• $\sqrt{15}$

• $\sqrt{17}$

• $\sqrt{18}$

If $\mathrm{a} = \hat{\mathrm{i}} +\hspace{0.17em}2\hat{\mathrm{j}} + 2\hat{\mathrm{k}}, \left|\mathrm{b}\right| = 5$ and the angle between a and b is $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right.$, then the area of the triangle formed by these two vectors as two sides is

• $\frac{15}{4}$

• $\frac{15}{2}$

• 15

• $\frac{15\sqrt{3}}{2}$

If a - b = 0 and a + b makes an angle of 60° with a, then

• $\left|\mathrm{a}\right| = 2\left|\mathrm{b}\right|$

• $2\left|\mathrm{a}\right| = \left|\mathrm{b}\right|$

• $\left|\mathrm{a}\right| = \sqrt{3}\left|\mathrm{b}\right|$

• $\sqrt{3}\left|\mathrm{a}\right| = \left|\mathrm{b}\right|$

If $\hat{\mathrm{i}} + \hat{\mathrm{j}}, \hat{\mathrm{j}} + \hat{\mathrm{k}}, \hat{\mathrm{i}} + \hat{\mathrm{k}}$ are the position vectors of the vertices of a $∆$ABC taken in order, then $\angle$A is equal to

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$5$}\right.$

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right.$

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$

Let a = $\hat{\mathrm{i}} - 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$. If b is a vector such that a · b = ${\left|\mathrm{b}\right|}^2$ and  $\left|\mathrm{a} - \mathrm{b}\right|$ = $\sqrt{7}$, then $\left|\mathrm{b}\right|$ is equal to

• $\sqrt{7}$

• $\sqrt{3}$

• 7

• 3

If a, b and c are three non-zero vectors such that each one of them being perpendicular to the sum of the other two vectors, then the value of ${\left|\mathrm{a} + \mathrm{b} + \mathrm{c}\right|}^2$ is

• ${\left|\mathrm{a}\right|}^2 + {\left|\mathrm{b}\right|}^2 + {\left|\mathrm{c}\right|}^2$

• $\left|\mathrm{a}\right| + \left|\mathrm{b}\right| + \left|\mathrm{c}\right|$

• 2$\left({\left|\mathrm{a}\right|}^2 + {\left|\mathrm{b}\right|}^2 + {\left|\mathrm{c}\right|}^2\right)$

• $\frac{1}{2}\left({\left|\mathrm{a}\right|}^2 + {\left|\mathrm{b}\right|}^2 + {\left|\mathrm{c}\right|}^2\right)$

Let u, v and w be vectors such that u + v + w = 0. If $\left|\mathrm{u}\right| = 3, \left|\mathrm{v}\right| = 4 \mathrm{and} \left|\mathrm{w}\right| = 5$, then u - v + v · w + w · u is equal to

• 0

• - 25

• 25

• 50

If $\mathrm{\lambda }\left(3\hat{\mathrm{i}} + 2\hat{\mathrm{j}} - 6\hat{\mathrm{k}}\right)$ is a unit vector, then the value of $\mathrm{\lambda }$ are

• $\pm \frac{1}{7}$

• $\pm 7$

• $\pm \sqrt{43}$

• $\pm \frac{1}{\sqrt{43}}$

119.

If the direction cosines of a vector of magnitude 3 are $\frac{2}{3}, \frac{- \mathrm{a}}{3}, \frac{2}{3}, \mathrm{a} >\hspace{0.17em}0$, then the vector is

• $2\hat{\mathrm{i}} + \hat{\mathrm{j}} + 2\hat{\mathrm{k}}$

• $2\hat{\mathrm{i}} - \hat{\mathrm{j}} + 2\hat{\mathrm{k}}$

   

• $\hat{\mathrm{i}} - 2\hat{\mathrm{j}} + 2\hat{\mathrm{k}}$

• $\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 2\hat{\mathrm{k}}$

Equation of the plane through the mid-point of the line segment joining the points P(4, 5, - 10), Q(- 1, 2, 1) and perpendicular to PQ is

• $\mathrm{r} . \left(\frac{3}{2}\hat{\mathrm{i}} + \frac{7}{2}\hat{\mathrm{j}} - \frac{9}{2}\hat{\mathrm{k}}\right) = 45$

• $\mathrm{r} . \left(- \hat{\mathrm{i}} + 2\hat{\mathrm{j}} + \hat{\mathrm{k}}\right) = \frac{135}{2}$

• $\mathrm{r} . \left(5\hat{\mathrm{i}} +\hspace{0.17em}3\hat{\mathrm{j}} - 11\hat{\mathrm{k}}\right) + \frac{135}{2} = 0$

• $\mathrm{r} . \left(5\hat{\mathrm{i}} + 3\hat{\mathrm{j}} - 11\hat{\mathrm{k}}\right) = \frac{135}{2}$