For any vectors, a, b, c $\mathrm{a} \times \left(\mathrm{b} + \mathrm{c}\right) + \mathrm{b} \times \left(\mathrm{c} + \mathrm{a}\right) + \mathrm{c} \times \left(\mathrm{a} +\hspace{0.17em}\mathrm{b}\right)$ = ?

• 0

• a + b + c

• [a, b, c]

• $\mathrm{a} \times \mathrm{b} \times \mathrm{c}$

If $\mathrm{a} = \hat{\mathrm{i}} + 4\hat{\mathrm{j}}, \mathrm{b} = 2\hat{\mathrm{i}} - 2\hat{\mathrm{j}}, \mathrm{c} = 5\hat{\mathrm{i}} + 9\hat{\mathrm{j}}$, then c is equal to

• 2a + b

• a + 2b

• 3a + b

• a + 3b

If $\mathrm{a} = \hat{\mathrm{i}} + \hat{\mathrm{j}} + \mathrm{t}\hat{\mathrm{k}}, \mathrm{b} = \hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$ then the values of 't' for which (a + b) and (a - b) are perpendicular, are

• $\pm 2$

• $\pm 2\sqrt{3}$

• $\pm 3\sqrt{2}$

• $\pm 3$

$\left[\hat{\mathrm{i}} - \hat{\mathrm{j}}, \hat{\mathrm{j}} - \hat{\mathrm{k}}, \hat{\mathrm{k}} - \hat{\mathrm{i}}\right]$ is equal to

• 0

• 1

• 2

• 3

405.

$\mathrm{If} \mathrm{a} . \hat{\mathrm{i}} = \mathrm{a} . \left(\hat{\mathrm{i}} + \hat{\mathrm{j}}\right) = \mathrm{a}\left(\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}\right), \mathrm{then} \mathrm{a} \mathrm{is} \mathrm{equal} \mathrm{to}$

• $\hat{\mathrm{i}}$

• $\hat{\mathrm{j}}$

• $\hat{\mathrm{k}}$

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

If three points A,B,and C have position vectors $\left(1, \mathrm{x}, 3\right), \left(3, 4, 7\right) \mathrm{and} \left(\mathrm{y}, - 2, - 5\right)$ respectively and if they are collinear, then (x, y) is

• $(2, - 3)$

• ( - 2, 3)

• ( - 2, - 3)

• $(2, 3)$

The orthogonal projection of a on b is

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

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

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

• $\frac{\mathrm{b}}{\left|\mathrm{b}\right|}$

If the position vectors of the vertices of atriangle are $2\hat{\mathrm{i}} - \hat{\mathrm{j}} + \hat{\mathrm{k}}$, $\hat{\mathrm{i}} - 3\hat{\mathrm{j}} - 5\hat{\mathrm{k}}$ and $3\hat{\mathrm{i}} - 4\hat{\mathrm{j}} - 4\hat{\mathrm{k}}$, then the probability of triangle is

• equilateral

• isosceless

• right angled isosceless

• right angled

If [a b c] = 3, then the volume (in cubic units) of e parallelepiped with 2a + b, 2b + c and 2c + a as edges, is

• 15

• 22

• 25

• 27

$\left(\mathrm{a} +\hspace{0.17em}\mathrm{b}\right) . \left(\mathrm{b} +\hspace{0.17em}\mathrm{c}\right) \times \left(\mathrm{a} + \mathrm{b} +\hspace{0.17em}\mathrm{c}\right)$ is equal to

• 0

• - [a b c]

• 2[a b c]

• [a b c]