If the vectors 3i - 4j - k and 2i + 3j - 6k represent the diagonals of a rhombus, then the length of the side of the rhombus is

• 15

• $15\sqrt{3}$

• $\frac{5\sqrt{3}}{2}$

• $\frac{15\sqrt{3}}{2}$

If a = 2i + 3j + $\mathrm{\alpha }$k and b = 3i - $\mathrm{\alpha }$j + 2k, then the angle between a + b and a-b is equal to

• 0

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$

If a = 2i + 2j - k, b = $\mathrm{\alpha i} + \mathrm{\beta j}$ + 2k and $\left|\mathrm{a} + \mathrm{b}\right| = \left|\mathrm{a} - \mathrm{b}\right|$, then $\mathrm{\alpha } + \mathrm{\beta }$ is equal to

• 2

• 1

• 0

• - 1

If the projection of b on a is twice the projection of a on b, then $\left|\mathrm{b}\right| - \left|\mathrm{a}\right|$ is equal to 

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

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

• $\left|\mathrm{b}\right|$

• $\left|\mathrm{a}\right|$

If a = i - j and b = j + k, then then ${\left|\mathrm{a} \times \mathrm{b}\right|}^2 + {\left|\mathrm{a} . \mathrm{b}\right|}^2$ is equal to

• $\sqrt{2}$

• 2

• $\sqrt{6}$

• 4

If $\left|\mathrm{a}\right| = 1, \left|\mathrm{b}\right| = 3 \mathrm{and} \left|\mathrm{a} - \mathrm{b}\right| = \sqrt{7}$, then the angle between a and b is

• 0

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{3}$

37.

A vector of magnitude 7 units, parallel to the resultant of the vectors a = 2 i - 3j - 2k and b = - i + 2j + k, is

• $\frac{7}{\sqrt{3}}\left(\mathrm{i} +\hspace{0.17em}\mathrm{j}\hspace{0.17em}+\hspace{0.17em}\mathrm{k}\right)$

• 7(i - j - k)

• $\frac{7}{\sqrt{3}}\left(\mathrm{i} - \mathrm{j} +\hspace{0.17em}\mathrm{k}\right)$

• $\frac{7}{\sqrt{3}}\left(\mathrm{i} - \mathrm{j} - \mathrm{k}\right)$

The point which divides the line joining the points (1, 3, 4) and (4, 3, 1) internally in the ratio 2 : 1, is

• (2, - 3, 3)

• (2, 3, 3)

• $\left(\frac{5}{2}, 3, \frac{{\displaystyle 5}}{{\displaystyle 2}}\right)$

• (3, 3, 2)

The angle between the lines $\frac{\mathrm{x} - 7}{1} = \frac{\mathrm{y} + 3}{- 5} = \frac{\mathrm{z}}{3}$ and $\frac{2 - \mathrm{x}}{- 7} = \frac{\mathrm{y}}{2} = \frac{\mathrm{z} + 5}{1}$ is equal to

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{6}$

The equation of the plane which is equidistant from the two parallel planes 2x - 2y + z + 3= 0 and 4x - 4y + 2z + 9 = 0 is

• 8x - 8y + 4z + 15 = 0

• 8x - 8y + 4z - 15 = 0

• 8x - 8y + 4z + 3 = 0

• 8x - 8y + 4z - 3 = 0