If x, y and z are non-zero real numbers and $\hat{\mathrm{a}} = \mathrm{x}\hat{\mathrm{i}} + 2\hat{\mathrm{j}}$, $\hat{\mathrm{b}} = \mathrm{y}\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$ and $\hat{\mathrm{c}} = \mathrm{x}\hat{\mathrm{i}} + \mathrm{y}\hat{\mathrm{j}} + \mathrm{z}\hat{\mathrm{k}}$ are such that $\hat{\mathrm{a}} \times \hat{\mathrm{b}} = \mathrm{z}\hat{\mathrm{i}} - 3\hat{\mathrm{j}} + \hat{\mathrm{k}}$, then $\left[\hat{\mathrm{a}} \hat{\mathrm{b}} \hat{\mathrm{c}}\right]$ equals to

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If M₁, M₂, M₃ and M₄ are respectiyely the magnitudes of _the vectors a₁ = 2i - j + k, a₂ = - 3i - 4j - 4k, a₃ = - i + j - k, a₄ = - i + 3j + k, then the correct order of M₁, M₂, M₃ and M₄ is

• M₃ < M₁ < M₄ < M₂

• M₃ < M₁ < M₂ < M₄

• M₃ < M₄ < M₁ < M₂

• M₃ < M₄ < M₂ < M₁

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

• $\frac{3}{2}$

• _{$- \frac{3}{2}$}

• $\frac{1}{2}$

• $- \frac{1}{2}$

$\mathrm{If} \mathrm{a} = 2\hat{\mathrm{i}} +\hspace{0.17em}\hat{\mathrm{k}}, \mathrm{b} = \hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}, \mathrm{c} = 4\hat{\mathrm{i}} - 3\hat{\mathrm{j}} + 7\hat{\mathrm{k}}$, then the vector r satisfying r x b = c x b and r · a = 0 is

• $\hat{\mathrm{i}} + 8\hat{\mathrm{j}} +\hspace{0.17em}2\hat{\mathrm{k}}$

• $\hat{\mathrm{i}} - 8\hat{\mathrm{j}} +\hspace{0.17em}2\hat{\mathrm{k}}$

• $\hat{\mathrm{i}} - 8\hat{\mathrm{j}} -\hspace{0.17em}2\hat{\mathrm{k}}$

• $- \hat{\mathrm{i}} - 8\hat{\mathrm{j}} +\hspace{0.17em}2\hat{\mathrm{k}}$

If a, b and c are three vectors such that $\left|\mathrm{a}\right| = 1, \left|\mathrm{b}\right| = 2, \left|\mathrm{c}\right| = 3 \mathrm{and} \mathrm{a} . \mathrm{b} = \mathrm{b} . \mathrm{c} = \mathrm{c} . \mathrm{a} = 0, \mathrm{then} \left|\left[\mathrm{a} \mathrm{b} \mathrm{c}\right]\right| = ?$ is equal to

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

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817.

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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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

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

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

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

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

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