21.

The mid point of the line joining the points (- 10, 8) and (- 6, 12)divides the line joining the points ( 4, - 2) and (- 2, 4) in the ratio

• 1 : 2 internally

• 1 : 2 externally

• 2 : 1 internally

• 2 : 1 externally

The position vectors of the points A and B with respect to O are 2i + 2j + k and 2i + 4j+ 4k. The length of the internal bisector of $\angle$BOA of $∆$AOB is

• $\frac{\sqrt{136}}{9}$

• $\frac{\sqrt{136}}{3}$

• $\frac{20}{3}$

• $\frac{\sqrt{217}}{9}$

Given (a x b) x (c x d) = 5c x 6d, then the value of (a . b) x (a + c + 2d) is

• 7

• 16

• - 1

• 4

Let a, b and c be non-zero vectors such that $\left(\mathrm{a} \times \mathrm{b}\right) \times \mathrm{c} = \frac{- 1}{4}\left|\mathrm{b}\right|\left|\mathrm{c}\right|\mathrm{a}$.  If $\mathrm{\theta }$ the acute angle between the vectors b and c, then the angle between a and c is equal to

• $\frac{2\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

A vector of magnitude 12 unit perpendicular to the plane containing the vectors 4i + 6j - k and 3i + 8j + k is

• - 8i + 4j + 8k

• 8i + 4j + 8k

• 8i - 4j + 8k

• 8i - 4j - 8k

Forces of magnitudes 3 and 4 unit acting alon 6i + 2j + 3k and 3i - 2j + 6k, respectively act on a particle and displace it from (2, 2,- 1) to (4, 3, 1). The work done is

• $\frac{124}{7}$

• $\frac{120}{7}$

• $\frac{125}{7}$

• $\frac{121}{7}$

If ABCD be a parallelogram and M be the point of intersection of the diagonals. If O is any point, then OA + OB + OC + OD is

• 3OM

• 4OM

• OM

• 2OM

If D, E and F are the mid points of the sides BC, CA and AB, respectively of the $∆$ABC and G is the centroid of the triangle then GD + GE + GF is

• 0

• 2AB

• 2GA

• 2GC

If from a point P (a, b, c) perpendiculars PA, PB are drawn to yz and zx planes, then the equation of the plane OAB is

• bcx + cay + abz = 0

• bcx + cay - abz = 0

• bcx - cay + abz = 0

• - bcx + cay + abz = 0

If P (x, y, z) is a point on the line segment joning Q (2, 2, 4) and R (3, 5, 6) such thatprojections of OP on the axes are $\frac{13}{5}, \frac{19}{5}, \frac{26}{5}$ respectively, then P divides QR in the ratio

• 1 : 2

• 3 : 2

• 2 : 3

• 1 : 3