An unit vector perpendicular to both $\hat{\mathrm{i}} + \hat{\mathrm{j}} \mathrm{and} \hat{\mathrm{j}} + \hat{\mathrm{k}}$ is :

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

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

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

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

482.

The area of the triangle whose vertices are (1, 2, 3), (2, 5, -1) and (-1, 1, 2) is :

• 150 sq unit

• 145 sq unit

• $\frac{\sqrt{155}}{2} \mathrm{sq} \mathrm{unit}$

• $\frac{155}{2} \mathrm{sq} \mathrm{unit}$

For any three vectors $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$, $\overrightarrow{\mathrm{a}} \times \left(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}\right)$ + $\overrightarrow{\mathrm{b}} \times \left(\overrightarrow{\mathrm{c}} + \overrightarrow{\mathrm{a}}\right)$ + $\overrightarrow{\mathrm{c}} \times \left(\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}}\right)$ is :

• $\overrightarrow{0}$

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

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

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

If $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ are any three vectors, then $\left[\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}} \overrightarrow{\mathrm{c}} + \overrightarrow{\mathrm{a}}\right]$ is equal to

• $\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]$

• 0

• $2\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]$

• ${\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]}^2$

The volume of the parallelopiped whose coterminus edges are $\hat{\mathrm{i}} - \hat{\mathrm{j}} + \hat{\mathrm{k}}, 2\hat{\mathrm{i}} - 4\hat{\mathrm{j}} + 5\hat{\mathrm{k}} \mathrm{and} 3\hat{\mathrm{i}} - 5\hat{\mathrm{j}} + 2\hat{\mathrm{k}}$ is :

• 4 cu unit

• 3 cu unit

• 2 cu unit

• 8 cu unit

For any vector $\overrightarrow{\mathrm{a}}, \hat{\mathrm{i}} \times \left(\overrightarrow{\mathrm{a}} \times \hat{\mathrm{i}}\right) + \hat{\mathrm{j}} \times \left(\stackrel{\to }{\mathrm{a}}\times \hat{\mathrm{j}}\right) + \hat{\mathrm{k}} \times \left(\overrightarrow{\mathrm{a}} \times \hat{\mathrm{k}}\right)$ is equal to:

• $\overrightarrow{0}$

• $\overrightarrow{\mathrm{a}}$

• $2\overrightarrow{\mathrm{a}}$

• $3\overrightarrow{\mathrm{a}}$

If the vectors $3\hat{\mathrm{i}} + \mathrm{\lambda }\hat{\mathrm{j}} + \hat{\mathrm{k}} \mathrm{and} 2\hat{\mathrm{i}} - \hat{\mathrm{j}} + 8\hat{\mathrm{k}}$ are perpendicular, then $\mathrm{\lambda }$ is :

• - 14

• 7

• 14

• $\frac{1}{7}$

The projection of the vector $\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}$ along the vector $\hat{\mathrm{j}}$ is:

• 1

• 0

• 2

• - 1

The projection of $\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + \hat{\mathrm{k}}$ on $2\hat{\mathrm{i}} - 3\hat{\mathrm{j}} + 6\hat{\mathrm{k}}$ is :

• 1/7

• - 1/7

• 7

• - 7

If $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}} = 0$ and $\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{b}}$ = 0, then :

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

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

• $\overrightarrow{\mathrm{a}} = \overrightarrow{0} \mathrm{and} \overrightarrow{\mathrm{b}} = \overrightarrow{0}$

• $\overrightarrow{\mathrm{a}} = \overrightarrow{0} \mathrm{or} \overrightarrow{\mathrm{b}} = \overrightarrow{0}$