Mathematics : Vector Algebra

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

If the vectors $\overrightarrow{\mathrm{c}}, \overrightarrow{\mathrm{a}} = \mathrm{x}\hat{\mathrm{i}} + \mathrm{y}\hat{\mathrm{j}} + \mathrm{z}\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{b}} = \hat{\mathrm{j}}$ are such that $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{c}} \mathrm{and} \overrightarrow{\mathrm{b}}$ form a right handed system, then $\overrightarrow{\mathrm{c}}$ is

• $\mathrm{z}\hat{\mathrm{i}} - \mathrm{x}\hat{\mathrm{k}}$

• $\overrightarrow{0}$

• $\mathrm{y}\hat{\mathrm{j}}$

• $- \mathrm{z}\hat{\mathrm{i}} + \mathrm{x}\hat{\mathrm{k}}$

The vector $\hat{\mathrm{i}} + \mathrm{x}\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$ is rotated through an angle $\mathrm{\theta }$ and doubled in magnitude, then it becomes $4\hat{\mathrm{i}} + \left(4\mathrm{x} - 2\right)\hat{\mathrm{j}} + 2\hat{\mathrm{k}}$. Then, the values of x are

• $- \frac{2}{3}, 2$

• $\frac{1}{3}, 2$

• $\frac{2}{3}, 0$

• 2, 7

If a = (1, - 1) and b = (- 2, m) are two collinear vectors, then m is equal to

• 4

• 3

• 2

• 0

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

• 2a

• 3a

• - 2a

• a

If the a, band c form the sides BC, CA and AB respectively, of a $∆$ABC, then

• $\mathrm{a} . \mathrm{b} + \mathrm{b} . \mathrm{c} + \mathrm{c} . \mathrm{a} = 0$

• $\mathrm{a} \times \mathrm{b} = \mathrm{b} \times \mathrm{c} = \mathrm{c} \times \mathrm{a}$

• $\mathrm{a} . \mathrm{b} = \mathrm{b} . \mathrm{c} = \mathrm{c} . \mathrm{a} = 0$

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