If D, E and F are respectively the mid-points of AB, AC and BC in $∆\mathrm{ABC}$, then $\overrightarrow{\mathrm{BE}} + \overrightarrow{\mathrm{AF}}$ is equal to :

• $\overrightarrow{\mathrm{DC}}$

• $\frac{1}{2}\overrightarrow{\mathrm{BF}}$

• $2\overrightarrow{\mathrm{BF}}$

• $\frac{3}{2}\overrightarrow{\mathrm{BF}}$

Let $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ be the position vectors of the vertices A, B, C respectively of $∆$ABC. Thevector area of $∆$ABC is :

• $\frac{1}{2}\left(\overrightarrow{\mathrm{a}}\left(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}\right) + \overrightarrow{\mathrm{b}}\left(\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}}\right) + \overrightarrow{\mathrm{c}}\left(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right)\right)$

• $\frac{1}{2}\left(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}} + \overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}}\right)$

• $\frac{1}{2}\left(\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}\right)$

• $\frac{1}{2}\left(\overrightarrow{\mathrm{a}}\left(\overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right) + \overrightarrow{\mathrm{b}}\left(\overrightarrow{\mathrm{c}} \overrightarrow{\mathrm{a}}\right) + \overrightarrow{\mathrm{c}}\left(\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}}\right)\right)$

413.

If $\overrightarrow{\mathrm{a}} = \hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}$, $\overrightarrow{\mathrm{b}} = \hat{\mathrm{i}} + \hat{\mathrm{j}}$, $\overrightarrow{\mathrm{c}} = \hat{\mathrm{i}}$ and $\left(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right) \times \overrightarrow{\mathrm{c}} = \mathrm{\lambda }\overrightarrow{\mathrm{a}} + \mathrm{\mu }\overrightarrow{\mathrm{b}}$, then $\mathrm{\lambda } + \mathrm{\mu }$ is equal to :

• 0

• 1

• 2

• 3

$$\begin{gathered} \mathrm{If} \hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}}, 3\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + \hat{\mathrm{k}} \mathrm{are} \mathrm{sides} \mathrm{of} \mathrm{a} \mathrm{parallelogram}, \mathrm{then} \mathrm{a} \\ \mathrm{unit} \mathrm{vector} \mathrm{is} \mathrm{parallel} \mathrm{to} \mathrm{one} \mathrm{of} \mathrm{the} \mathrm{diagonals} \mathrm{of} \mathrm{the} \mathrm{parallelogram} \mathrm{is} \end{gathered}$$

• $\frac{\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}}{\sqrt{3}}$

• $\frac{\hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}}}{\sqrt{3}}$

• $\frac{\hat{\mathrm{i}} - \hat{\mathrm{j}} + \hat{\mathrm{k}}}{\sqrt{3}}$

• $\frac{- \hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}}{\sqrt{3}}$

If G is the centroid of the $∆$ABC, then GA + BG + GC is equal

• 2GB

• 2GA

• 0

• 2BG

If the vectors $\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + 4\hat{\mathrm{k}}, \mathrm{\lambda }\hat{\mathrm{i}} - 4\hat{\mathrm{j}} + \hat{\mathrm{k}}$ are orthogonal to each other, then $\mathrm{\lambda }$ is equal to

• 5

• - 5

• 8

• - 8

The vector c . (b + c) x (a + b + c) is equal to

• c . b x a

• 0

• c . a x b

• a . c x b

If the vector $\mathrm{a} = 2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + 6\hat{\mathrm{k}}$ and b are collinear and $\left|\mathrm{b}\right| = 21$, then b is equal to

• $\pm \left(2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + 6\hat{\mathrm{k}}\right)$

• $\pm 3\left(2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + 6\hat{\mathrm{k}}\right)$

• $\left(\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}\right)$

• $\pm 21\left(2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + 6\hat{\mathrm{k}}\right)$

If a and b are unit vectors, then the vector (a + b) x (a x b) is parallel to the vector

• a - b

• a + b

• 2a - b

• 2a + b

I. Two non-zero, non-collinear vectors arelinearly independent.

II. Any three coplanar vectors are linearlydependent.Which ofthe above statements is/are true?

• Only I

• Only II

• Both I and II

• Neither I nor II