The area ofthe parallelogram whose adjacent sides are $\hat{\mathrm{i}} + \hat{\mathrm{k}} \mathrm{and} 2\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}$ is

• 3

• $\sqrt{2}$

• 4

• $\sqrt{3}$

If a and b are two unit vectors mclined at an angle $\frac{\mathrm{\pi }}{3}$ then the value of $\left|\mathrm{a} + \mathrm{b}\right|$ is

• equal to

• greater than 1

• equal to 0

• less than 1

The value of [(a - b)(b - c)(c - a)] is equal to

• 0

• 1

• 2[a b c]

• 2

Let a = $\hat{\mathrm{i}} - 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$, if b is a vector such that $\mathrm{a} . \mathrm{b} = {\left|\mathrm{b}\right|}^2$ and $\left|\mathrm{a} - \mathrm{b}\right| = \sqrt{7}, \mathrm{then} \left|\mathrm{b}\right|$, then lbl is equal to

• $\sqrt{7}$

• 7

• 21

• 14

Let a = $\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + \hat{\mathrm{k}}$, b = $\hat{\mathrm{i}} - \hat{\mathrm{j}} + \hat{\mathrm{k}}$ and c = $\hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}}$. A vector in the plane a and b whose projection on c is $\frac{1}{\sqrt{3}}$, is

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

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

• $4\hat{\mathrm{i}} - \hat{\mathrm{j}} + 4\hat{\mathrm{k}}$

• $4\hat{\mathrm{i}} + \hat{\mathrm{j}} - 4\hat{\mathrm{k}}$

If a and b are unit vectors, then what is the angle between a and b for  $\sqrt{3}\mathrm{a} - \mathrm{b}$ to be unit vector?

• $30°$

• $45°$

• $60°$

• 90$°$

If a = 3, b = 4, c = 5 each one of a, b and c is perpendicular to the sum of the remaining, then $\left|\mathrm{a} + \mathrm{b} + \mathrm{c}\right|$ is equal to

• $\frac{5}{\sqrt{2}}$

• $\frac{2}{\sqrt{5}}$

• $5\sqrt{2}$

• $\sqrt{5}$

The value of x, if x$\left(\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}\right)$ is a unit vector, is

• $\pm \frac{1}{\sqrt{3}}$

• $\pm \sqrt{3}$

• $\pm 3$

• $\pm \frac{1}{3}$

If a $= 2\hat{\mathrm{i}} + \mathrm{\lambda }\hat{\mathrm{j}} + \hat{\mathrm{k}}$ and b = $\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$ are orthogonal, then value of $\mathrm{\lambda }$ is

• 3/2

• 1

• 0

• - 5/2

If a, b, c are unit vectors such that a + b + c = 0, then the value of a · b + b · c + c · a is equal to

• 3/2

• 1

• 3

• - 3/2