If $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}} = \overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{d}} \mathrm{and} \overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{c}} = \overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{d}}$, then

• $\left(\overrightarrow{\mathrm{a}} - \overrightarrow{\mathrm{d}}\right) = \mathrm{\lambda }\left(\overrightarrow{\mathrm{b}} - \overrightarrow{\mathrm{c}}\right)$

• $\left(\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{d}}\right) = \mathrm{\lambda }\left(\overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}\right)$

• $\left(\overrightarrow{\mathrm{a}} - \overrightarrow{\mathrm{b}}\right) = \mathrm{\lambda }\left(\overrightarrow{\mathrm{c}} + \overrightarrow{\mathrm{d}}\right)$

• $\left(\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}}\right) = \mathrm{\lambda }\left(\overrightarrow{\mathrm{c}} - \overrightarrow{\mathrm{d}}\right)$

A unit vector in xy-plane makes an angle of 45° with the vector $\hat{\mathrm{i}} + \hat{\mathrm{j}}$ and an angle of 60° with the vector $3\hat{\mathrm{i}} - 4\hat{\mathrm{j}}$, is

• $\hat{\mathrm{i}}$

• $\frac{\hat{\mathrm{i}} + \hat{\mathrm{j}}}{\sqrt{2}}$

• $\frac{\hat{\mathrm{i}} - \hat{\mathrm{j}}}{\sqrt{2}}$

• None of these

Let a, b, c be distinct non-negative numbers. If the vector $\mathrm{a}\hat{\mathrm{i}} + \mathrm{a}\hat{\mathrm{j}}+ \mathrm{c}\hat{\mathrm{k}}, \hat{\mathrm{i}}+ \hat{\mathrm{k}} \mathrm{and} \mathrm{c}\hat{\mathrm{i}} + \mathrm{c}\hat{\mathrm{j}} + \mathrm{b}\hat{\mathrm{k}}$, i+f and ci + cj + bk lies in a plane, then c is

• the GM of a and b

• the GM of a and b

• the HM of a and b

• equal to zero

If $\overrightarrow{\mathrm{p}} \times \overrightarrow{\mathrm{q}} = \overrightarrow{\mathrm{r}} \mathrm{and} \overrightarrow{\mathrm{q}} \times \overrightarrow{\mathrm{r}} = \overrightarrow{\mathrm{p}}$, then

• r = 1, p = q

• p = 1, q = 1

• r = 2p, q = 2

• q = 1, p = r

Vectors $\overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}}$ are inclined at an angle 8 = 120°. If $\left|\overrightarrow{\mathrm{a}}\right| = 1 \mathrm{and} \left|\overrightarrow{\mathrm{b}}\right| = 2,$ then ${\left[\left(\overrightarrow{\mathrm{a}} + 3\overrightarrow{\mathrm{b}}\right) \times \left(3\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}}\right)\right]}^2$ is equal to 

• 190

• 275

• 300

• 192

426.

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.$

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