Write the direction cosines of the vector  $- 2 \hat{\mathrm{i}} + \hat{\mathrm{j}} - 5 \mathrm{k}.$

Find a unit vector perpendicular to each of the vector  $\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{a}} - \overrightarrow{\mathrm{b}}$, where

$\overrightarrow{\mathrm{a}} = 3 \hat{\mathrm{i}} + 2 \hat{\mathrm{j}} + 2 \hat{\mathrm{k}} \mathrm{and} \overrightarrow{\mathrm{b}} = \hat{\mathrm{i}} + 2 \hat{\mathrm{j}} - 2 \hat{\mathrm{k}}.$

Find ' $\mathrm{\lambda }$ ' when the projection of  $\overrightarrow{\mathrm{a}} = \mathrm{\lambda } \hat{\mathrm{i}} + \hat{\mathrm{j}} + 4 \hat{\mathrm{k}} \mathrm{on} \overrightarrow{\mathrm{b}} = 2 \hat{\mathrm{i}} + 6 \hat{\mathrm{j}} + 3 \hat{\mathrm{k}}$ is  4  units.

Find the sum of the following vectors: 

$\overrightarrow{\mathrm{a}} = \hat{\mathrm{i}} - 2 \hat{\mathrm{j}} , \overrightarrow{\mathrm{b}} = 2 \hat{\mathrm{i}} - 3 \hat{\mathrm{j}} , \overrightarrow{\mathrm{c}} =2 \hat{\mathrm{i}} + 3 \hat{\mathrm{k}}.$

If  $\overrightarrow{\mathrm{a}} , \overrightarrow{\mathrm{b}} , \overrightarrow{\mathrm{c}}$  are three vectors such that $\left| \overrightarrow{\mathrm{a}} \right| = 5, \left| \overrightarrow{\mathrm{b}} \right| = 12$  and $\left| \overrightarrow{\mathrm{c}} \right| = 13 \mathrm{and} \overrightarrow{a} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}} = 0$  find the value of  $\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{b}} . \overrightarrow{\mathrm{c}} + \overrightarrow{\mathrm{c}} . \overrightarrow{\mathrm{a}} .$

The Boolean Expression (p∧~q)∨q∨(~p∧q) is equivalent to:

• ~p ∧ q

• p ∧ q

• p ∨ q

• p ∨ q

347.

Let, a, b and c be three non-zero vectors such that no two of them are collinear and  if θ is the angle between vectors b and  c, then a value of sin θ is

• 
• 
• 
• 

If the vectors AB = 3î + 4k̂ and AC = 5î - 2ĵ + 4k̂ are the sides of a Δ ABC, then the length of the median through A is 

• √18

• √72

• √33

• √45

Let  be  two unit vectors. If the vectors  and  are perpendicular to each other, then the angle between  is 

• π/6

• π/2

• π/3

• π/3

Let ABCD be a parallelogram such that  and ∠BAD be an acute angle. If   is the vector that coincides with the altitude directed from the vertex B the side AD, then  is given byLet ABCD be a parallelogram such that AB = q,AD = p and ∠BAD be an acute angle. If r is the vector that coincides with the altitude directed from the vertex B to the side AD, then r is given by (1)