31.

If the vector 8i + aj of magnitude 10 is in the direction of the vector 4i + 3j, then the value of equal to

• 6

• 3

• - 3

• - 6

If a = 2i - 7j + k and b = i + 3j - 5k and a · mb = 120, then the value of m is equal to

• 5

• - 24

• - 5

• 120

If the angle between a and c is 25°, the angle between b and c is 65° and a + b = c, then the angle between a and b is

• 40°

• 115°

• 25°

• 90°

The position vector of the centroid of the $∆$ABC is 2i + 4 j + 2k. If the position vector of the vertex A is 2i + 6j + 4k, then the position vector of midpoint of BC is

• 2i + 3j + k

• 2i + 3j - k

• 2i - 3j - k

• - 2i - 3j - k

The projection of the vector 2i + aj - k on the vector i - 2j + k is $\frac{- 5}{\sqrt{6}}$. Then, the value of a is equal to

• 1

• 2

• - 2

• 3

A unit vector in the XOY-plane that makes an angle 30° with the vector i + j and makes an angle 60° with i - j is

• $\frac{1}{4}\left[\left(\sqrt{6} + \sqrt{2}\right)\mathrm{i} - \left(\sqrt{6} - \sqrt{2}\right)\mathrm{j}\right]$

• $\frac{1}{2}\left[\left(\sqrt{6} - \sqrt{2}\right)\mathrm{i} + \left(\sqrt{6} + \sqrt{2}\right)\mathrm{j}\right]$

• $\frac{1}{4}\left[\left(\sqrt{6} - \sqrt{2}\right)\mathrm{i} + \left(\sqrt{6} + \sqrt{2}\right)\mathrm{j}\right]$

• $\frac{1}{4}\left[\left(\sqrt{6} + \sqrt{2}\right)\mathrm{i} + \left(\sqrt{6} - \sqrt{2}\right)\mathrm{j}\right]$

The angle between the line r = (i + 2j + 3k) + $\mathrm{\lambda }$(2i + 3j + 4k) and the plane r - (i + j - 2k) = 0 is

• 0°

• 60°

• 30°

• 90°

The lines r = i + j - k + (3i - j) and r = 4j - k + µ (2i + 3k) intersect at the point

• (0, 0, 0)

• (0, 0, 1)

• (0, - 4, - 1)

• (4, 0, - 1)

An equation of the plane through the points (1, 0, 0) and (0, 2, 0) and at a distance $\frac{6}{7}$ units from the origin is

• 6x + 3y + z - 6 = 0

• 6x + 3y + 2z - 6= 0

• 6x + 3y + z + 6 = 0

• 6x + 3y + 2z + 6 = 0

The projection of a line segment on the axes are 9, 12 and 8. Then, the length of the line segment is

• 15

• 16

• 17

• 18