If 2i + 3j, i + j + k and $\mathrm{\lambda }$i + 4j + 2k taken in an order are coterminous edges of a parallelopiped of volume 2 cu units, then value of $\mathrm{\lambda }$ is

• - 4

• 2

• 3

• 4

A unit vector perpendicular to both i + j + k and 2i + j + 3k is

• $\left(2\mathrm{i} - \mathrm{j} - \mathrm{k}\right)\sqrt{6}$

• $\frac{\left(2\mathrm{i} - \mathrm{j} - \mathrm{k}\right)}{\sqrt{6}}$

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

• $\frac{\left(3\mathrm{i} + \mathrm{j} - 2\mathrm{k}\right)}{\sqrt{6}}$

If a, b and c are three non-coplanar vectors and p, q and r are vectors defined by $\mathrm{p} = \frac{\mathrm{b} \times \mathrm{c}}{\left[\mathrm{a} \mathrm{b} \mathrm{c}\right]}, \mathrm{q} = \frac{\mathrm{c} \times \mathrm{a}}{\left[\mathrm{a} \mathrm{b} \mathrm{c}\right]} \mathrm{and} \mathrm{r} = \frac{\mathrm{a} \times \mathrm{b}}{\left[\mathrm{a} \mathrm{b} \mathrm{c}\right]}$, then the value of (a + b) . p + (b + c) . q + (c + a) . r is equal to

• 0

• 1

• 2

• 3

If (a $\times$ b)² + (a . b)² = 144 and $\left|\mathrm{a}\right|$ = 4, $\left|\mathrm{b}\right|$ is equal to

• 16

• 8

• 3

• 12

If i + j - k and 2i - 3j + k are adjacent sides of a parallelogram, then the lengths of its diagonals are

• $\sqrt{3}, \sqrt{14}$

• $\sqrt{13}, \sqrt{14}$

• $\sqrt{21}, \sqrt{3}$

• $\sqrt{21}, \sqrt{13}$

If the volume of the parallelepiped formed by three non-coplanar vectors a, b and c is 4 cu units, then [a x b b x c c x a] is equal to

• 64

• 16

• 4

• 8

If a = (1,2, 3), b = (2, - 1, 1), c = (3, 2, 1) and $\mathrm{a} \times \left(\mathrm{b} \times \mathrm{c}\right) = \mathrm{\alpha a} + \mathrm{\beta b} + \mathrm{\gamma c}$, then

• $\mathrm{\alpha } = 1, \mathrm{\beta } = 10, \mathrm{\gamma } = 3$

• $\mathrm{\alpha } = 0, \mathrm{\beta } = 10, \mathrm{\gamma } = - 3$

• $\mathrm{\alpha } + \mathrm{\beta } + \mathrm{\gamma } = 8$

• $\mathrm{\alpha } = \mathrm{\beta } = \mathrm{\gamma } = 0$

If $\mathrm{a} \perp \mathrm{b} \mathrm{and} \left(\mathrm{a} + \mathrm{b}\right) \perp \left(\mathrm{a} + \mathrm{mb}\right)$, then m is equal to

• - 1

• 1

• $\frac{- {\left|\mathrm{a}\right|}^2}{{\left|\mathrm{b}\right|}^2}$

• 0

If a, b and c are unit vectors such that a + b + c = 0, then a . b + b . c + c . a is equal to

• $\frac{3}{2}$

• $- \frac{3}{2}$

• $\frac{2}{3}$

• $\frac{1}{2}$

If a is a vector perpendicular to both b and c, then

• $\mathrm{a} . \left(\mathrm{b} \times \mathrm{c}\right) = 0$

• $\mathrm{a} \times \left(\mathrm{b} \times \mathrm{c}\right) = 0$

• $\mathrm{a} \times \left(\mathrm{b} + \mathrm{c}\right) = 0$

• $\mathrm{a} + \left(\mathrm{b} + \mathrm{c}\right) = 0$