Volume of the parallelopiped having vertices at 0 = (0, 0, 0), A = (2, - 2, 1), B = (5, - 4, 4) and C = (1, - 2, 4) is

• 5 cu unit

• 10 cu unit

• 15 cu unit

• 20 cu unit

542.

If $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ are three non-coplanar vectors and $\overrightarrow{\mathrm{p}}, \overrightarrow{\mathrm{q}}, \overrightarrow{\mathrm{r}}$ are defined by the relations

$\overrightarrow{\mathrm{p}} = \frac{\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}}{\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]}, \overrightarrow{\mathrm{q}} = \frac{\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}}}{\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]} \mathrm{and} \overrightarrow{\mathrm{r}} = \frac{\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{a}}}{\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]}$

then $\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{p}} + \overrightarrow{\mathrm{b}} . \overrightarrow{\mathrm{q}} + \overrightarrow{\mathrm{c}} . \overrightarrow{\mathrm{r}}$ is equl to

• 0

• 1

• 2

• 3

The volume of a parallelopiped whose coterminous edges are  $2\overrightarrow{\mathrm{a}}, \overrightarrow{2\mathrm{b}}, 2\overrightarrow{\mathrm{c}}$ is

• $2\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]$

• $4\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]$

• $8\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]$

• $\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]$

The position vectors of vertices of a $∆$ABC are $4\hat{\mathrm{i}} - 2\hat{\mathrm{j}}$, $\hat{\mathrm{i}} + 4\hat{\mathrm{j}} - 3\hat{\mathrm{k}}$ and $- \hat{\mathrm{i}} + 5\hat{\mathrm{j}}+ \hat{\mathrm{k}}$ respectively, then $\angle$ABC is equal to

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

If u = a - b and v = a + b and $\left|\mathrm{a}\right| = \left|\mathrm{b}\right|$ = 2, then $\left|\mathrm{u} \times \mathrm{v}\right|$ is equal to

• $2\sqrt{16 - {\left(\mathrm{a} . \mathrm{b}\right)}^2}$

• $\sqrt{16 - {\left(\mathrm{a} . \mathrm{b}\right)}^2}$

• $\mathrm{2}\sqrt{4 - {\left(\mathrm{a} . \mathrm{b}\right)}^2}$

• $2\sqrt{4 + {\left(\mathrm{a} . \mathrm{b}\right)}^2}$

If the vectors a, b and c are coplanar, then

$\left|\begin{array}{ccc}\mathrm{a}& \mathrm{b}& \mathrm{c}\\ \mathrm{a} . \mathrm{a}& \mathrm{a} . \mathrm{b}& \mathrm{a} . \mathrm{c}\\ \mathrm{b} . \mathrm{a}& \mathrm{b} . \mathrm{b}& \mathrm{b} . \mathrm{c}\end{array}\right|$ is equal to

• 1

• 0

• - 1

• None of these

If the position vectors of the vertices A, B and C are 6i, 6j and k respectively w.r.t. origin 0, then the volume of the tetrahedron OABC is

• 6

• 3

• $\frac{1}{6}$

• $\frac{1}{3}$

If three vectors 2i - j - k, i + 2j - 3k and 3i + $\mathrm{\lambda }$j + 5k are coplanar, then the value of $\mathrm{\lambda }$ is

• - 4

• - 2

• - 1

• - 8

The vector perpendicular to the vectors 4i - j + 3k and - 2i + j - 2k whose magnitude is 9

• 3i + 6j - 6k

• 3i - 6j + 6k

• - 3i + 6j + 6k

• None of the above

$\mathrm{i} . \left(\mathrm{j} \times \mathrm{k}\right) + \mathrm{j} . \left(\mathrm{k} \times \mathrm{i}\right) + \mathrm{k} . \left(\mathrm{j} \times \mathrm{i}\right)$ is equal to

• 3

• 2

• 1

• 0