The projection of the vector i - 2j + k on the vector 4i - 4j + 7k is

• $\frac{5\sqrt{6}}{10}$

• $\frac{19}{9}$

• $\frac{9}{19}$

• $\frac{\sqrt{6}}{19}$

If a, b, c are three non-zero vectors such that a + b+ c = 0 and m = a . b + b . c + c · a, then

• m < 0

• m > 0

• m = 0

• m = 3

If $\overrightarrow{\mathrm{a}} = 2\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$, $\overrightarrow{\mathrm{b}} = - \hat{\mathrm{i}} + 2\hat{\mathrm{j}} + \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{c}} = 3\hat{\mathrm{i}} + \hat{\mathrm{j}}$ then $\overrightarrow{\mathrm{a}} + \mathrm{t}\overrightarrow{\mathrm{b}}$ is perpendicular to c, if t is equal to

• 2

• 4

• 6

• 8

404.

The distance between the line $\overrightarrow{\mathrm{r}} = 2\hat{\mathrm{i}} - 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}} + \mathrm{\lambda }\left(\hat{\mathrm{i}} - \hat{\mathrm{j}} + 4\hat{\mathrm{k}}\right)$ and the plane $\overrightarrow{\mathrm{r}} . \left( \hat{\mathrm{i}} + 5\hat{\mathrm{j}} + \hat{\mathrm{k}}\right) = 5$, is

• $\frac{10}{3}$

• $\frac{10}{\sqrt{3}}$

• $\frac{10}{3\sqrt{3}}$

• $\frac{10}{9}$

If m₁, m₂, m₃ and m₄ are respectively the magnitudes of the vectors

$$\begin{gathered} \overrightarrow{{\mathrm{a}}_1} = 2\hat{\mathrm{i}} - \hat{\mathrm{j}} + \hat{\mathrm{k}}, \overrightarrow{{\mathrm{a}}_2} = 3\hat{\mathrm{i}} - 4\hat{\mathrm{j}} - 4\hat{\mathrm{k}}, \\ \overrightarrow{{\mathrm{a}}_3} = \hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}} \mathrm{and} \overrightarrow{{\mathrm{a}}_4} = - \hat{\mathrm{i}} + 3\hat{\mathrm{j}} + \hat{\mathrm{k}}, \end{gathered}$$

then the correct order of m₁, m₂, m₃ and m₄ is

• m₃ < m₁ < m₄ < m₂

• m₃ < m₁ < m₂ < m₄

• m₃ < m₄ < m₁ < m₂

• m₃ < m₄ < m₂ < m₁

If $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ be three unit vectors such that $\overrightarrow{\mathrm{a}} \times \left(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}\right) = \frac{1}{2}\overrightarrow{\mathrm{b}}$, $\overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{c}}$ being non-parallel. If ${\mathrm{\theta }}_1$ is the angle between $\overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}}$ and ${\mathrm{\theta }}_2$ is the angle between $\overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{c}}$, then

• ${\mathrm{\theta }}_1 = \frac{\mathrm{\pi }}{6}, {\mathrm{\theta }}_2 = \frac{\mathrm{\pi }}{3}$

• ${\mathrm{\theta }}_1 = \frac{\mathrm{\pi }}{3}, {\mathrm{\theta }}_2 = \frac{\mathrm{\pi }}{6}$

• ${\mathrm{\theta }}_1 = \frac{\mathrm{\pi }}{2}, {\mathrm{\theta }}_2 = \frac{\mathrm{\pi }}{3}$

• ${\mathrm{\theta }}_1 = \frac{\mathrm{\pi }}{3}, {\mathrm{\theta }}_2 = \frac{\mathrm{\pi }}{2}$

The ${\overrightarrow{\mathrm{r}}}^2 - \overrightarrow{\mathrm{r}} . \overrightarrow{\mathrm{c}} + \mathrm{h} = 0$, $\left|\overrightarrow{\mathrm{c}}\right| > \sqrt{\mathrm{h}}$, represents

• circle

• ellipse

• cone

• sphere

$\overrightarrow{\mathrm{a}} = \hat{\mathrm{i}} - \hat{\mathrm{j}} + \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{b}} = 2\hat{\mathrm{i}} + 4\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$ are one of the sides and medians respectively, of a triangle through the same vertex, then area of the triangle is

• $\frac{1}{2}\sqrt{83}$

• $\sqrt{83}$

• $\frac{1}{2}\sqrt{85}$

• $\sqrt{86}$

If $\overrightarrow{\mathrm{b}}$ is a unit vector, then $\left(\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{b}}\right)\overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{b}}\left(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right)$ is :

• ${\left|\overrightarrow{\mathrm{a}}\right|}^2 \overrightarrow{\mathrm{b}}$

• $\left|\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{b}}\right|\overrightarrow{\mathrm{a}}$

• $\overrightarrow{\mathrm{a}}$

• $\overrightarrow{\mathrm{b}}$

The position vectors of the points A and B with respect to O are 2i + 2j + k and 2i + 4j+ 4k. The length of the internal bisector of $\angle$BOA of $∆$AOB is

• $\frac{\sqrt{136}}{9}$

• $\frac{\sqrt{136}}{3}$

• $\frac{20}{3}$

• $\frac{\sqrt{217}}{9}$