Mathematics : Vector Algebra

$$\begin{gathered} \mathrm{Let} \overrightarrow{\mathrm{a}} \mathrm{be} \mathrm{a} \mathrm{unit} \mathrm{vector}, \overrightarrow{\mathrm{b}} = 2\hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}} \mathrm{and} \overrightarrow{\mathrm{c}} =\hat{\mathrm{i}} + 3\hat{\mathrm{k}}. \\ \mathrm{Then}, \mathrm{maximum} \mathrm{value} \mathrm{of}[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b} }\overrightarrow{\mathrm{c}}] \mathrm{is} \end{gathered}$$

• - 1

• $\sqrt{10} + \sqrt{6}$

• $\sqrt{10} - \sqrt{6}$

• $\sqrt{59}$

If $\overrightarrow{\mathrm{a}} = \hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}, \overrightarrow{\mathrm{b}} = \hat{\mathrm{i}} - \hat{\mathrm{j}} + \hat{\mathrm{k}}, \overrightarrow{\mathrm{c}} = \hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{d}} =\hat{\mathrm{i}} - \hat{\mathrm{j}} - \hat{\mathrm{k}}$, then observe the following lists

	List-I		List-II
(i)	$\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{b}}$	(A)	$\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{d}}$
(ii)	$\overrightarrow{\mathrm{b}} . \overrightarrow{\mathrm{c}}$	(B)	3
(iii)	$\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]$	(C)	$\overrightarrow{\mathrm{b}} . \overrightarrow{\mathrm{d}}$
(iv)	$\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}$	(D)	$2\hat{\mathrm{i}} - \hat{\mathrm{k}}$
		(E)	$2\hat{\mathrm{j}} + 2\hat{\mathrm{k}}$
		(F)	4

Then, correct match of List_I to List-II is

$$\begin{gathered} \mathrm{If} {\mathrm{m}}_1, {\mathrm{m}}_2, {\mathrm{m}}_3 \mathrm{and} {\mathrm{m}}_4 \mathrm{are} \mathrm{respectively} \mathrm{the} \mathrm{magnitudes} \mathrm{of} \mathrm{the} \mathrm{vectors} \\ {\overrightarrow{\mathrm{a}}}_1 = 2\hat{\mathrm{i}} - \hat{\mathrm{j}} + \hat{\mathrm{k}}, {\overrightarrow{\mathrm{a}}}_2 = 3\hat{\mathrm{i}} - 4\hat{\mathrm{j}} - 4\hat{\mathrm{k}}, \\ {\overrightarrow{\mathrm{a}}}_3 = \hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}} \mathrm{and} {\overrightarrow{\mathrm{a}}}_4 = - \hat{\mathrm{i}} + 3\hat{\mathrm{j}} + \hat{\mathrm{k}}, \mathrm{then} \mathrm{the} \mathrm{correct} \mathrm{order} \mathrm{of} {\mathrm{m}}_1, {\mathrm{m}}_2, {\mathrm{m}}_3 \mathrm{and} {\mathrm{m}}_4 \mathrm{is} \end{gathered}$$

• ${\mathrm{m}}_3 < {\mathrm{m}}_1 < {\mathrm{m}}_4 < {\mathrm{m}}_2$

• ${\mathrm{m}}_3 < {\mathrm{m}}_1 < {\mathrm{m}}_2 < {\mathrm{m}}_4$

• ${\mathrm{m}}_3 < {\mathrm{m}}_4 < {\mathrm{m}}_1 < {\mathrm{m}}_2$

• ${\mathrm{m}}_3 < {\mathrm{m}}_4 < {\mathrm{m}}_2 < {\mathrm{m}}_1$

$\mathrm{Let} \overrightarrow{\mathrm{a}} = \hat{\mathrm{i}} - 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}}, \overrightarrow{\mathrm{b}} = 2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} - \hat{\mathrm{k}}, \overrightarrow{\mathrm{c}} = \hat{\mathrm{i}} - 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$. If $\overrightarrow{\mathrm{c}}$ is parallel to the plane containing $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}$, then $\mathrm{\lambda }$ is equal to

• 0

• 1

• - 1

• 2

If three unit vectors $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ satisfy $\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}$, then the angle between a and b is

• $\frac{2\mathrm{\pi }}{3}$

• $\frac{5\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{6}$

$\left(\overrightarrow{\mathrm{a}} + \overrightarrow{2\mathrm{b}} - \overrightarrow{\mathrm{c}}\right) . \left(\overrightarrow{\mathrm{a}} - \overrightarrow{\mathrm{b}}\right) \times \left(\overrightarrow{\mathrm{a}} - \overrightarrow{\mathrm{b}} - \overrightarrow{\mathrm{c}}\right) = ?$

• $- \left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]$

• $2\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]$

• $3\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]$

$\mathrm{If} \overrightarrow{\mathrm{u}} = \overrightarrow{\mathrm{a}} - \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{v}} = \overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}}, \left|\overrightarrow{\mathrm{a}}\right| = \left|\overrightarrow{\mathrm{b}}\right| = 2, \mathrm{then} \left|\overrightarrow{\mathrm{u}} \times \overrightarrow{\mathrm{v}}\right| = ?$

• $2\sqrt{16 - {\left(\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{b}}\right)}^2}$

• $2\sqrt{16 + {\left(\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}}\right)}^2 }$

• $2\sqrt{4 - {\left(\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}}\right)}^2 }$

• $2\sqrt{4 + {\left(\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}}\right)}^2 }$

$$\begin{gathered} \mathrm{If} \mathrm{the} \mathrm{angle} \mathrm{\theta } \mathrm{between} \mathrm{the} \mathrm{vectors} \overrightarrow{\mathrm{a}} = 2{\mathrm{x}}^2\hat{\mathrm{i}} + 4\mathrm{x}\hat{\mathrm{j}} + \hat{\mathrm{k}} \mathrm{and} \overrightarrow{\mathrm{b}} = 7\hat{\mathrm{i}} - 2\hat{\mathrm{j}} +\mathrm{x}\hat{\mathrm{k}} \\ \mathrm{is} \mathrm{such} \mathrm{that} 90° <\mathrm{\theta } < 180°, \mathrm{then} \mathrm{x} \mathrm{lies} \mathrm{in} \mathrm{the} \mathrm{interval} \end{gathered}$$

• $\left(0, \frac{1}{2}\right)$

• $\left(\frac{1}{2}, 1\right)$

• $\left(1, \frac{3}{2}\right)$

• $\left(\frac{1}{2}, \frac{3}{2}\right)$

Let OA, OB, QC be the co-terminal edges of a rectangular parallelopiped of volume V and let P be the vertex opposite to O. Then, $\left[\overrightarrow{\mathrm{AP}} \overrightarrow{\mathrm{BP}} \overrightarrow{\mathrm{CP}}\right]$ is equal to

• 2V

• 12V

• $3\sqrt{3}\mathrm{V}$

• 0

If the vectors ii - 2xj - 3yk and i + 3xj + 2yk are orthogonal to each other, then the locus of the point (x, y) is

• a circle

• an ellipse

• a parabola

• a straight line