Mathematics : Vector Algebra

If a = $- \hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}$ and b = $2\hat{\mathrm{i}} + \hat{\mathrm{k}}$, then the vector satisfyin the following conditions

(i) it is coplanar witha and b,

(ii) it is perpendicular to b and

(iii) a · c = 7, is

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

• $- \frac{3}{2}\hat{\mathrm{i}} + \frac{5}{2}\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$

• $- 3\hat{\mathrm{i}} + 5\hat{\mathrm{j}} + 6\hat{\mathrm{k}}$

• $- 6\hat{\mathrm{i}} + \hat{\mathrm{k}}$

If the vectors $\mathrm{b} = \left(\tan \left(\mathrm{\alpha }\right), - 1, 2\sqrt{\sin \left(\frac{\mathrm{\alpha }}{2}\right)}\right)$ and $\mathrm{c} = \left(\tan \left(\mathrm{\alpha }\right), \tan \left(\mathrm{\alpha }\right), - \frac{3}{\sqrt{\sin \left({\displaystyle \raisebox{1ex}{$\mathrm{\alpha }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)}}\right)$ are orthagonal and  a vector a = $\left(1, 3, \sin \left(2\mathrm{\alpha }\right)\right)$ makes an obtuse angle with the Z-axis, then the value of $\mathrm{\alpha }$ is

• $\left(4\mathrm{n} + 2\right)\mathrm{\pi } + {\tan }^{-1}\left(2\right)$

• $\left(4\mathrm{n} + 2\right)\mathrm{\pi } - {\tan }^{-1}\left(2\right)$

• $\left(4\mathrm{n} + 1\right)\mathrm{\pi } + {\tan }^{-1}\left(2\right)$

• $\left(4\mathrm{n} + 1\right)\mathrm{\pi } - {\tan }^{-1}\left(2\right)$

${\left(\mathrm{r} . \hat{\mathrm{i}}\right)}^2 + {\left(\mathrm{r} . \hat{\mathrm{j}}\right)}^2 + {\left(\mathrm{r} . \hat{\mathrm{k}}\right)}^2$ is equal to

• 0

• 1

• r²

• 3r²

The component of $\hat{\mathrm{i}} + \hat{\mathrm{j}}$ along $\hat{\mathrm{j}} \mathrm{and} \hat{\mathrm{k}}$ will be

• $\frac{\hat{\mathrm{i}} + \hat{\mathrm{j}}}{2}$

• $\frac{\hat{\mathrm{j}} + \hat{\mathrm{k}}}{2}$

• $\frac{\hat{\mathrm{k}} + \hat{\mathrm{i}}}{2}$

• None of these

If $\mathrm{a} = 2\hat{\mathrm{i}} + 5\hat{\mathrm{j}}$ and $\mathrm{b} = 2\hat{\mathrm{i}} - \hat{\mathrm{j}}$, then the unit vector along a + b will be

• $\frac{\hat{\mathrm{i}} - \hat{\mathrm{j}}}{\sqrt{2}}$

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

• $\sqrt{2}\left(\hat{\mathrm{i}} + \hat{\mathrm{j}}\right)$

• $\frac{\hat{\mathrm{i}} + \hat{\mathrm{j}}}{\sqrt{2}}$

For any vectors, a, b, c $\mathrm{a} \times \left(\mathrm{b} + \mathrm{c}\right) + \mathrm{b} \times \left(\mathrm{c} + \mathrm{a}\right) + \mathrm{c} \times \left(\mathrm{a} +\hspace{0.17em}\mathrm{b}\right)$ = ?

• 0

• a + b + c

• [a, b, c]

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

If $\mathrm{a} = \hat{\mathrm{i}} + 4\hat{\mathrm{j}}, \mathrm{b} = 2\hat{\mathrm{i}} - 2\hat{\mathrm{j}}, \mathrm{c} = 5\hat{\mathrm{i}} + 9\hat{\mathrm{j}}$, then c is equal to

• 2a + b

• a + 2b

• 3a + b

• a + 3b

If $\mathrm{a} = \hat{\mathrm{i}} + \hat{\mathrm{j}} + \mathrm{t}\hat{\mathrm{k}}, \mathrm{b} = \hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$ then the values of 't' for which (a + b) and (a - b) are perpendicular, are

• $\pm 2$

• $\pm 2\sqrt{3}$

• $\pm 3\sqrt{2}$

• $\pm 3$

$\left[\hat{\mathrm{i}} - \hat{\mathrm{j}}, \hat{\mathrm{j}} - \hat{\mathrm{k}}, \hat{\mathrm{k}} - \hat{\mathrm{i}}\right]$ is equal to

• 0

• 1

• 2

• 3

$\mathrm{If} \mathrm{a} . \hat{\mathrm{i}} = \mathrm{a} . \left(\hat{\mathrm{i}} + \hat{\mathrm{j}}\right) = \mathrm{a}\left(\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}\right), \mathrm{then} \mathrm{a} \mathrm{is} \mathrm{equal} \mathrm{to}$

• $\hat{\mathrm{i}}$

• $\hat{\mathrm{j}}$

• $\hat{\mathrm{k}}$

• $\left(\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}\right)$