If the projection of the vector $\overrightarrow{\mathrm{a}} \mathrm{on} \overrightarrow{\mathrm{b}}$ is $\left|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right|$ and if $3\overrightarrow{\mathrm{b}} = \hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}$, then the angle between $\overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}}$ is

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

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

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$

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

82.

If $\overrightarrow{\mathrm{x}} = \overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{y}} = \overrightarrow{\mathrm{a}} - \overrightarrow{\mathrm{b}}, \left|\overrightarrow{\mathrm{a}}\right| = 2, \left|\overrightarrow{\mathrm{b}}\right| = 3$ and  the angle between $\overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}}$ is $\frac{\mathrm{\pi }}{3}$, then $\left|\overrightarrow{\mathrm{x}} \times \overrightarrow{\mathrm{y}}\right|$ is equal to

• $5\sqrt{3}$

• 6

• $4\sqrt{3}$

• $6\sqrt{3}$

If the position vectors of three consecutive vertices of a parallelogram are $\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}$, $\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + 5\hat{\mathrm{k}}$ and $7\hat{\mathrm{i}} + 9\hat{\mathrm{j}} + 11\hat{\mathrm{k}}$, then the coordinates of the fourth vertex are

• (2, 1, 3)

• (6, 7, 8)

• (4, 1, 3)

• (7, 7, 7)

The two variable vectors $3\mathrm{x} \hat{\mathrm{i}} + \mathrm{y} \hat{\mathrm{j}} + - 3\hat{\mathrm{k}}$ and $\mathrm{x} \mathrm{i} - 4\mathrm{y} \hat{\mathrm{j}} + 4\hat{\mathrm{k}}$ are orthogonal to each other, then the locus of (x, y) is

• hyperbola

• circle

• straight line

• ellipse

If $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ are non-coplanar and $\left(\overrightarrow{\mathrm{a}} \times \mathrm{\lambda }\overrightarrow{\mathrm{b}}\right) . \left[\left(\overrightarrow{\mathrm{b}} \times 3\overrightarrow{\mathrm{c}}\right) \times \left(\overrightarrow{\mathrm{c}} - 4\overrightarrow{\mathrm{a}}\right)\right]$ = 0, then the value of $\mathrm{\lambda }$ is equal to

• 0

• $\frac{1}{12}$

• $\frac{5}{12}$

• 3

The vector form of the sphere 2(x² + y² + z²) - 4x + 6y + 8z - 5 = 0 is

• $\overrightarrow{\mathrm{r}} . \left[\overrightarrow{\mathrm{r}} - \left(2\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}\right)\right] = \frac{2}{5}$

• $\overrightarrow{\mathrm{r}} . \left[\overrightarrow{\mathrm{r}} - \left(2\hat{\mathrm{i}} - 3\hat{\mathrm{j}} - 4\hat{\mathrm{k}}\right)\right] = \frac{1}{2}$

• $\overrightarrow{\mathrm{r}} . \left[\overrightarrow{\mathrm{r}} - \left(2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + 4\hat{\mathrm{k}}\right)\right] = \frac{5}{2}$

• $\overrightarrow{\mathrm{r}} . \left[\overrightarrow{\mathrm{r}} - \left(2\hat{\mathrm{i}} - 3\hat{\mathrm{j}} - 4\hat{\mathrm{k}}\right)\right] = \frac{5}{2}$

If $\left|\overrightarrow{\mathrm{a}}\right| = 5, \left|\overrightarrow{\mathrm{b}}\right| = 6 \mathrm{and} \overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{b}} = - 25, \mathrm{then} \left|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right| \mathrm{is} \mathrm{equal} \mathrm{to}$

• 25

• $6\sqrt{11}$

• 11$\sqrt{5}$

• $5\sqrt{11}$

If $\overrightarrow{\mathrm{p}}, \overrightarrow{\mathrm{q}} \mathrm{and} \overrightarrow{\mathrm{r}}$ are perpendicular to $\overrightarrow{\mathrm{q}} + \overrightarrow{\mathrm{r}},$ $\overrightarrow{\mathrm{r}} + \overrightarrow{\mathrm{p}},$ $\overrightarrow{\mathrm{p}} + \overrightarrow{\mathrm{q}}$ respectively and if $\left|\overrightarrow{\mathrm{p}} + \overrightarrow{\mathrm{q}}\right| = 6,$ $\left|\overrightarrow{\mathrm{q}} + \overrightarrow{\mathrm{r}}\right|= 4\sqrt{3}$ and $\left|\overrightarrow{\mathrm{r}} + \overrightarrow{\mathrm{p}}\right| = 4,$  then $\left|\overrightarrow{\mathrm{p}} + \overrightarrow{\mathrm{q}} + \overrightarrow{\mathrm{r}}\right|$ is

• 5$\sqrt{2}$

• 10

• 15

• 5

The vectors of magnitude a, 2a, 3a meet at a point and their directions are along the diagonals of three adjacent faces of a cube. Then, the magnitude of their resultant is

• 5a

• 6a

• 10a

• 9a

Which one of the following vectors is of magnitude 6 and perpendicular to both $\overrightarrow{\mathrm{a}} = 2\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + \hat{\mathrm{k}}$ $\mathrm{and} \overrightarrow{\mathrm{b}} = \hat{\mathrm{i}} - 2\hat{\mathrm{j}} + 2\hat{\mathrm{k}}$ ?

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

• $2\left(2\hat{\mathrm{i}} - \hat{\mathrm{j}} + 2\hat{\mathrm{k}}\right)$

• $3\left(2\hat{\mathrm{i}} - \hat{\mathrm{j}} - 2\hat{\mathrm{k}}\right)$

• $2\left(2\hat{\mathrm{i}} - \hat{\mathrm{j}} - 2\hat{\mathrm{k}}\right)$