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

If a = $\hat{\mathrm{i}} + \hat{\mathrm{j}} - 2\hat{\mathrm{k}}, \mathrm{b} = 2\hat{\mathrm{i}} - \hat{\mathrm{j}} + \hat{\mathrm{k}}$ and c = $3\hat{\mathrm{i}} - \hat{\mathrm{k}}$ and c = ma + nb, then m + n is equal to

• 0

• 1

• 2

• - 1

M and N are the mid-points of the diagonals AC and BO respectively of quadrilateral ABCD, then AB + AD + CB + CD is equal to

• 2MN

• 2NM

• 4MN

• 4NM

If $\mathrm{a} = \hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}$, $\mathrm{b} = 2\hat{\mathrm{i}} + \mathrm{\lambda }\hat{\mathrm{j}} + \hat{\mathrm{k}}$, $\mathrm{c} = \hat{\mathrm{i}} - \hat{\mathrm{j}} + 4\hat{\mathrm{k}}$ and $\mathrm{a} . \left(\mathrm{b} \times \mathrm{c}\right)$ = 10, then $\mathrm{\lambda }$ is equal to

• 6

• 7

• 9

• 10

Let $\square$PQRS be a quadrilateral. If M and N are the mid-points of the sides PQ and RS respectively, then PS + QR =

• 3 MN

• 4 MN

• 2 MN

• 2 NM

If vector r with dc's l, m, n is equally inclined to the coordinate axes, then the total number of such vectors is

• 4

• 6

• 8

• 2