The volume ofa parallelepiped whose sides are given by a = $2\hat{\mathrm{i}} - 3\hat{\mathrm{j}}$, b = $\hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}}$ and c = $3\hat{\mathrm{i}} - \hat{\mathrm{k}}$ is

• 6 cu unit

• 5 cu unit

• 4 cu unit

• 3 cu unit

If a, b, c are non-coplanar vectors and 11, is a real number, then the vectors a + 2b + 3c, $\mathrm{\lambda }$b + 4c and (2$\mathrm{\lambda }$ - 1)c are non - coplanar for

• all values of $\mathrm{\lambda }$

• all except one value of $\mathrm{\lambda }$

• all expect two values of $\mathrm{\lambda }$

• no value of $\mathrm{\lambda }$

If u = $2\hat{\mathrm{i}} + 2\hat{\mathrm{j}} - \hat{\mathrm{k}}$ and v = $6\hat{\mathrm{i}} - 3\hat{\mathrm{j}} + 2\hat{\mathrm{k}}$, then the unit vector perpendicular to u and v is

• $\hat{\mathrm{i}} - 10\hat{\mathrm{j}} - 18\hat{\mathrm{k}}$

• $\frac{1}{\sqrt{17}}\left(\frac{1}{5}\hat{\mathrm{i}} - 2\hat{\mathrm{j}} - \frac{18}{5}\hat{\mathrm{k}}\right)$

• $\frac{1}{\sqrt{473}}\left(7\hat{\mathrm{i}} - 10\hat{\mathrm{j}} - 18\hat{\mathrm{k}}\right)$

• None of the above

If a and b are antiparallel, then a · b is equal to

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

• $- \left|\mathrm{a}\right|\left|\mathrm{b}\right|$

• 0

• None of these

If the position vectors ofthe points A and B are $\hat{\mathrm{i}} + 3\hat{\mathrm{j}} - \hat{\mathrm{k}}$ and $3\hat{\mathrm{i}} - \hat{\mathrm{j}} - 3\hat{\mathrm{k}}$ respectively, then the position vector of the mid-point of AB is

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

• $2\hat{\mathrm{i}} + \hat{\mathrm{j}} - 2\hat{\mathrm{k}}$

• $2\hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}}$

• $\hat{\mathrm{i}} + \hat{\mathrm{j}} - 2\hat{\mathrm{k}}$

366.

If the vectors $\mathrm{a}\hat{\mathrm{i}} - 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$ and $3\hat{\mathrm{i}} + 6\hat{\mathrm{j}} - 5\hat{\mathrm{k}}$ are perpendicularto each other, then a is given by

• 9

• 16

• 25

• 36

If position vectors of points A,B,C are $\hat{\mathrm{i}}, \hat{\mathrm{j}}, \hat{\mathrm{k}}$ respectively and AB = CX, then position vector of point X is

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

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

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

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

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