The angle between the straight lines $\frac{\mathrm{x} + 1}{2} = \frac{\mathrm{y} - 2}{5} = \frac{\mathrm{z} + 3}{4}$ and $\frac{\mathrm{x} - 1}{1} = \frac{\mathrm{y} + 2}{2} = \frac{\mathrm{z} - 3}{- 3}$ is

• $45°$

• $30°$

• $60°$

• $90°$

The acute angle between the planes 2x - y + z = 6 and x + y + 2z = 3 is

• $45°$

• $60°$

• $30°$

• $90°$

63.

The ratio in which the line joining the points (2, 4, 5) and (3, 5, - 4) is divided by the YZ-plane, is

• 2 : 3

• 3 : 2

• - 2 : 3

• 4 : - 3

If the position yectors of the points A, B, C, D are $2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + 5\hat{\mathrm{k}}$, $\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$, $- 5\hat{\mathrm{i}} + 4\hat{\mathrm{j}} - 2\hat{\mathrm{k}}$ and $\hat{\mathrm{i}} + 10\hat{\mathrm{j}} + 10\hat{\mathrm{k}}$, then

• AB = CD

• $\mathrm{AB} \parallel \mathrm{CD}$

• $\mathrm{AB} \perp \mathrm{CD}$

• None of the above

If $\left|\mathrm{a}\right| = 3, \left|\mathrm{b}\right| = 4, \left|\mathrm{c}\right| = 5$ are such that each of them is perpendicular to the sum ofthe other two, then $\left|\mathrm{a} + \mathrm{b} + \mathrm{c}\right|$ is equal to

• $5\sqrt{2}$

• $\frac{5}{\sqrt{2}}$

• $10\sqrt{2}$

• $5\sqrt{3}$

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

• $\frac{1}{\sqrt{14}}$

• $\frac{2}{\sqrt{14}}$

• $\frac{3}{\sqrt{14}}$

• None of these

If a and b are the position vectors of A and B, then position vector of point C on AB produced such that AC = 3AB, is

• 3a - b

• 3b - a

• 3a - 2b

• 3b - 2a

The area of a parallelogram whose adjacent sides are $\hat{\mathrm{i}} - 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$ and $2\hat{\mathrm{i}} + \hat{\mathrm{j}} - 4\hat{\mathrm{k}}$, is

• $5\sqrt{3}$

• $10\sqrt{3}$

• $5\sqrt{6}$

• $10\sqrt{6}$

${\left|\mathrm{a} \times \mathrm{b}\right|}^2 + {\left|\mathrm{a} . \mathrm{b}\right|}^2$ is equal to

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

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

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

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

A unit vector perpendicular to the plane of $\mathrm{a} = 2\hat{\mathrm{i}} - 6\hat{\mathrm{j}} - 3\hat{\mathrm{k}}$, $\mathrm{b} = 4\hat{\mathrm{i}} + 3\hat{\mathrm{j}} - \hat{\mathrm{k}}$, is

• $\frac{4\hat{\mathrm{i}} + 3\hat{\mathrm{j}} - \hat{\mathrm{k}}}{\sqrt{26}}$

• $\frac{2\hat{\mathrm{i}} - 6\hat{\mathrm{j}} - 6\hat{\mathrm{k}}}{7}$

• $\frac{3\hat{\mathrm{i}} - 2\hat{\mathrm{j}} + 6\hat{\mathrm{k}}}{7}$

• $\frac{2\hat{\mathrm{i}} - 3\hat{\mathrm{j}} - 6\hat{\mathrm{k}}}{7}$