Vector Algebra

$\mathrm{If} \left[\mathrm{a} \times \mathrm{b} \mathrm{b} \times \mathrm{c} \mathrm{c} \times \mathrm{a}\right] = \mathrm{\lambda }{\left[\mathrm{a} \mathrm{b} \mathrm{c}\right]}^2, \mathrm{then} \mathrm{\lambda } \mathrm{is} \mathrm{equal} \mathrm{to}$

• 0

• 1

• 2

• 3

The cartesion equation of the plane passing through the point (3, - 2, - 1) and parallel to the vectors $\mathrm{b} = \hat{\mathrm{i}} - 2\hat{\mathrm{j}} + 4\hat{\mathrm{k}} \mathrm{and} \mathrm{c} = 3\hat{\mathrm{i}} + 2\hat{\mathrm{j}} - 5\hat{\mathrm{k}} \mathrm{is}$

• 2x - 17y - 8z + 63 = 0 

• 3x + 17y + 8z + 36 = 0

• 2x + 17y + 8z + 36 = 0

• 3x - 16y + 8z - 63 = 0

$$\begin{gathered} \mathrm{If} \left|{\mathrm{z}}_1\right| = 1, \left|{\mathrm{z}}_2\right| = 2, \left|{\mathrm{z}}_3\right| = 3 \mathrm{and} \left|9{\mathrm{z}}_1{\mathrm{z}}_2 + 4{\mathrm{z}}_1{\mathrm{z}}_3 + {\mathrm{z}}_2{\mathrm{z}}_3\right| = 12, \mathrm{then} \\ \mathrm{the} \mathrm{value} \mathrm{of} \left|{\mathrm{z}}_1 +{\mathrm{z}}_2 + {\mathrm{z}}_3\right| \mathrm{is} \end{gathered}$$

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• 8

• 2

The cartesian equation of the plane whose vector equation is $\mathrm{\gamma } = \left(1 + \mathrm{\lambda } - \mathrm{\mu }\right)\hat{\mathrm{i}} + \left(2 - \mathrm{\lambda }\right)\hat{\mathrm{j}} + \left(3 - 2\mathrm{\lambda } + 2\mathrm{\mu }\right)\hat{\mathrm{k}}, \mathrm{where} \mathrm{\lambda }, \mathrm{\mu } \mathrm{are} \mathrm{scalars}, \mathrm{is}$ 

• 2x + y = 5

• 2x - y = 5

• 2x - z = 5

• 2x + z = 5

$\mathrm{For} \mathrm{three} \mathrm{vectors} \mathrm{p}, \mathrm{q} \mathrm{and} \mathrm{r}, \mathrm{if} \mathrm{r} = 3\mathrm{p} +\hspace{0.17em}4\mathrm{q} \mathrm{and} 2\mathrm{r} = \mathrm{p} - 3\mathrm{q}, \mathrm{then}$

• $\left|\mathrm{r}\right| < 2\left|\mathrm{q}\right| \mathrm{and} \mathrm{r}, \mathrm{q} \mathrm{have} \mathrm{the} \mathrm{same} \mathrm{direction}$

• $\left|\mathrm{r}\right| > 2\left|\mathrm{q}\right| \mathrm{and} \mathrm{r}, \mathrm{q} \mathrm{have} \mathrm{opposite} \mathrm{directions}$

• $\left|\mathrm{r}\right| < 2\left|\mathrm{q}\right| \mathrm{and} \mathrm{r}, \mathrm{q} \mathrm{have} \mathrm{opposite} \mathrm{directions}$

• $\left|\mathrm{r}\right| > 2\left|\mathrm{q}\right| \mathrm{and} \mathrm{r}, \mathrm{q} \mathrm{have} \mathrm{same} \mathrm{directions}$

$\mathrm{If} \mathrm{a} = 2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} - 5\hat{\mathrm{k}}, \mathrm{b} = \mathrm{m}\hat{\mathrm{i}} + \mathrm{n}\hat{\mathrm{j}} + 12\hat{\mathrm{k}} \mathrm{and} \mathrm{a} \times \mathrm{b} = 0, \mathrm{then} \left(\mathrm{m}, \mathrm{n}\right) = ?$

• $\left(\frac{- 24}{5}, \frac{- 36}{5}\right)$

• $\left(\frac{ - 24}{5}, \frac{36}{5}\right)$

• $\left(\frac{24}{5}, \frac{- 36}{5}\right)$

• $\left(\frac{24}{5}, \frac{36}{5}\right)$

$\mathrm{If} \left|\mathrm{a}\right| = 3, \left|\mathrm{b}\right| = 4 \mathrm{and} \mathrm{the} \mathrm{angle} \mathrm{between} \mathrm{a} \mathrm{and} \mathrm{b} \mathrm{is} 120°, \mathrm{then} \left|4\mathrm{a} +\hspace{0.17em}3\mathrm{b}\right| = ?$

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$$\begin{gathered} \mathrm{If} \mathrm{a}, \mathrm{b}, \mathrm{c} \mathrm{are} \mathrm{non}-\mathrm{zero} \mathrm{vectors} \mathrm{such} \mathrm{that} \left(\mathrm{a} \times \mathrm{b}\right) \times \mathrm{c} = \frac{1}{3}\left|\mathrm{b}\right|\left|\mathrm{c}\right|\mathrm{a}, \mathrm{c} ┴ \mathrm{a} \mathrm{and} \mathrm{\theta } \mathrm{is} \mathrm{the} \\ \mathrm{angle} \mathrm{between} \mathrm{the} \mathrm{vectors} \mathrm{b} \mathrm{and} \mathrm{c}, \mathrm{then} \sin \left(\mathrm{\theta }\right) = ? \end{gathered}$$

• $\frac{2\sqrt{2}}{3}$

• $\frac{1}{3}$

• $\frac{\sqrt{2}}{3}$

• $\frac{2}{3}$

$$\begin{gathered} \mathrm{If} \mathrm{\alpha }\left(\mathrm{\alpha } \times \mathrm{\beta }\right) + \mathrm{b}\left(\mathrm{\beta } \times \mathrm{\gamma }\right) + \mathrm{c}\left(\mathrm{\gamma } \times \mathrm{\alpha }\right) = 0 \mathrm{and} \mathrm{atleast} \mathrm{one} \mathrm{of} \mathrm{the} \mathrm{scalars} \mathrm{a},\mathrm{b}, \mathrm{c} \mathrm{is} \mathrm{non}-\mathrm{zero}, \\ \mathrm{then} \mathrm{the} \mathrm{vectors} \mathrm{\alpha }, \mathrm{\beta }, \mathrm{\gamma } \mathrm{are} \end{gathered}$$

• parallel

• non coplanar

• coplanar

• mutually perpendicular

If a non-zero vector a is parallel to the line of intersection of the plane determined by the vectors $\hat{\mathrm{j}} - \hat{\mathrm{k}}, 3\hat{\mathrm{j}} - 2\hat{\mathrm{k}}$ the plane determined by the vectors $2\hat{\mathrm{i}} + 3\hat{\mathrm{j}}, \widehat{\mathrm{i} }- 3\hat{\mathrm{j}}$ then the angle between the vectors a and $\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}$  is

• ${\sin }^{-1}\left(\frac{2}{\sqrt{3}}\right)$

• ${\cos }^{-1}\left(\pm \frac{2}{\sqrt{3}}\right)$

• ${\tan }^{-1}\left(\sqrt{3}\right)$

• ${\cos }^{-1}\left( \pm \frac{1}{\sqrt{3}}\right)$