Mathematics : Vector Algebra

The vector of magnitude 9 unit perpendicular to the vectors $4\hat{\mathrm{i}} - \hat{\mathrm{j}} + 3\hat{\mathrm{k}}$ and $- 2\hat{\mathrm{i}} + \hat{\mathrm{j}} - 2\hat{\mathrm{k}}$ will be

• $3\hat{\mathrm{i}} + 6\hat{\mathrm{j}} - 6\hat{\mathrm{k}}$

• $- 3\hat{\mathrm{i}} + 6\hat{\mathrm{j}} + 6\hat{\mathrm{k}}$

• $3\hat{\mathrm{i}} - 6\hat{\mathrm{j}} + 6\hat{\mathrm{k}}$

• $3\hat{\mathrm{i}} + 6\hat{\mathrm{j}} + 6\hat{\mathrm{k}}$

If $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}} = \overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{b}} \ne \overrightarrow{0}$, then $\overrightarrow{\mathrm{a}} - \overrightarrow{\mathrm{c}}$ will be equal to

• $\mathrm{k}\overrightarrow{\mathrm{b}}$

• $\mathrm{k}\overrightarrow{\mathrm{a}}$

• $\mathrm{k}\overrightarrow{\mathrm{c}}$

• $\mathrm{k}\left(\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}}\right)$

Which one of the following is not defined?

• $\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}}$

• $\overrightarrow{\mathrm{a}} \times 2\overrightarrow{\mathrm{b}}$

• $\left(3\overrightarrow{\mathrm{a}} \times 2\overrightarrow{\mathrm{b}}\right)4\overrightarrow{\mathrm{a}}$

• $\left(\overrightarrow{\mathrm{a}} . 2\overrightarrow{\mathrm{b}}\right)\overrightarrow{\mathrm{a}}$

The vectors $2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} - 4\hat{\mathrm{k}}$ and $\mathrm{a}\hat{\mathrm{i}} + \mathrm{b}\hat{\mathrm{j}} + \mathrm{c}\hat{\mathrm{k}}$ are perpendicular, when

• a = 2, b = 3, c = - 4

• a = 4, b = 4, c = 5

• a = 4, b = 4, c = - 5

• None of the above

A man runs due North at the rate of 20 km/h and the wind blows fromthe East at the rate of 15km/h. Then, the velocity of the wind relative to the man will be

• 2km/h

• 3 km/h

• 25 km/h

• None of these

The scalar A · {(B + C) x (A + B + C)} equals

• [ABC][BCA]

• [ABC]

• 0

• None of these

The points with position vectors 60i + 3j, 40i - 8 j and ai - 52j are collinear, if

• a = - 40

• a = 40

• a = - 20

• a = 20

If $\mathrm{\theta }$ is the angle between vectors p = ai + bj + ck and q = bi + cj + ak, then $\mathrm{\theta }$ lies in

• $\left[0, \frac{\mathrm{\pi }}{2}\right]$

• $\left[\frac{\mathrm{\pi }}{2}, \mathrm{\pi }\right]$

• $\left[\frac{\mathrm{\pi }}{2}, \frac{2\mathrm{\pi }}{3}\right]$

• $\left[0, \frac{2\mathrm{\pi }}{3}\right]$

If a, b and c are non-coplanar vectors and $\mathrm{p} = \frac{\mathrm{b} \times \mathrm{c}}{\left[\mathrm{a} \mathrm{b} \mathrm{c}\right]}$, q = $\mathrm{q} = \frac{\mathrm{c} \times \mathrm{a}}{\left[\mathrm{a} \mathrm{b} \mathrm{c}\right]}$ and $\mathrm{r} = \frac{\mathrm{a} \times \mathrm{b}}{\left[\mathrm{a} \mathrm{b} \mathrm{c}\right]}$, then a . p + b . q + c . r is equal to

• 3

• - 3

• 0

• None of the above

If a = 2i - 3j + 6k and b = - 2i + 2i - k, then $\frac{\mathrm{Projection} \mathrm{of} \mathrm{a} \mathrm{on} \mathrm{b}}{\mathrm{Projection} \mathrm{of} \mathrm{b} \mathrm{on} \mathrm{a}}$ is equal to

• 1

• 7/3

• 3/7

• - 1/6