The scalar A · {(B + C) x (A + B + C)} equals

• [ABC][BCA]

• [ABC]

• 0

• None of these

The points with position vectors 60i + 3j, 40i - 8 j and ai - 52j are collinear, if

• a = - 40

• a = 40

• a = - 20

• a = 20

If $\mathrm{\theta }$ is the angle between vectors p = ai + bj + ck and q = bi + cj + ak, then $\mathrm{\theta }$ lies in

• $\left[0, \frac{\mathrm{\pi }}{2}\right]$

• $\left[\frac{\mathrm{\pi }}{2}, \mathrm{\pi }\right]$

• $\left[\frac{\mathrm{\pi }}{2}, \frac{2\mathrm{\pi }}{3}\right]$

• $\left[0, \frac{2\mathrm{\pi }}{3}\right]$

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

• 3

• - 3

• 0

• None of the above

If a = 2i - 3j + 6k and b = - 2i + 2i - k, then $\frac{\mathrm{Projection} \mathrm{of} \mathrm{a} \mathrm{on} \mathrm{b}}{\mathrm{Projection} \mathrm{of} \mathrm{b} \mathrm{on} \mathrm{a}}$ is equal to

• 1

• 7/3

• 3/7

• - 1/6

The value of $\left(\mathrm{i} + \mathrm{j}\right) . \left[\left(\mathrm{j} + \mathrm{k}\right) \times \left(\mathrm{k} + \mathrm{i}\right)\right]$ is

• 0

• 1

• - 1

• 2

387.

If $\hat{\mathrm{a}} \mathrm{and} \hat{\mathrm{b}}$ are unit vectors and 0 is the angle between them, then $\sin \left(\raisebox{1ex}{$\mathrm{\theta }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)$ is equal to

• $\frac{\left|\hat{\mathrm{a}} + \hat{\mathrm{b}}\right|}{2}$

• $\frac{\hat{\mathrm{a}} - \hat{\mathrm{b}}}{2}$

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

• $\left|\hat{\mathrm{a}} - \hat{\mathrm{b}}\right|$

If a, b and c are three non-zero, non-coplanar vectors, then the value of a x a' + b x b'+ c x c' is

• 1

• 0

• - 1

• None of the above

Three concurrent edges of a parallelopiped are given by

$$\begin{gathered} \mathrm{a} = 2\hat{\mathrm{i}} - 3\hat{\mathrm{j}} + \hat{\mathrm{k}} \\ \mathrm{b} = \hat{\mathrm{i}} - \hat{\mathrm{j}} + 2\hat{\mathrm{k}} \\ \mathrm{and} \mathrm{c} = 2\hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}} \end{gathered}$$

The volume of the parallelopiped is

• 14 cu units

• 20 cu units

• 25 cu units

• 60 cu units

The number of vectors of unit length perpendicular to vectors a = $\hat{\mathrm{i}} + \hat{\mathrm{j}}$ and b = $\hat{\mathrm{j}} + \hat{\mathrm{k}}$

• infinite

• one

• two

• three