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

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{3}$

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)

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

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

If p and q are non-collinear unit vectors and $\left|\mathrm{p} + \mathrm{q}\right| = \sqrt{3}$, then (2p - 3q) · (3p + q) is equal to

• 0

• $\frac{1}{3}$

• $- \frac{1}{3}$

• $- \frac{1}{2}$

107.

The triangle formed by the three points whose position vectors are 2i + 4j - k, 4i + 5j + k and 3i + 6j - 3k, is

• an equilateral triangle

• a right angled triangle but not isosceles

• an isosceles triangle but not right angled triangle

• a right angled isosceles triangle

If (1, 2, 4) and (2, - 3$\mathrm{\lambda }$, - 3) are the initial and terminal points of the vector i + 5j - 7k, then the value of $\mathrm{\lambda }$ is

• $\frac{7}{3}$

• $\frac{- 7}{3}$

• $\frac{- 5}{3}$

• $\frac{5}{3}$

Let u = 5a + 6b + 7c, v = 7a - 8b + 9c and w = 3 a + 20b + 5c, where a, b and c are non-zero vectors. If u = lv + mw, then the values of l and m respectively are

• $\frac{1}{2}, \frac{1}{2}$

• $\frac{1}{2}, - \frac{1}{2}$

• $- \frac{1}{2}, \frac{1}{2}$

• $\frac{1}{3}, \frac{1}{3}$

If $\overrightarrow{\mathrm{\alpha }} = 3\mathrm{i} - \mathrm{k}$ and $\left|\overrightarrow{\mathrm{\beta }}\right| = \sqrt{5}$ and $\overrightarrow{\mathrm{\alpha }} . \overrightarrow{\mathrm{\beta }}$ = 3 then the area of the parallelogram for which $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$ are adjacent sides, is

• $\frac{\sqrt{17}}{2}$

• $\frac{\sqrt{14}}{2}$

• $\frac{\sqrt{7}}{2}$

• $\sqrt{41}$