A unit vector coplanar with i + j + 3k and i + 3j + k and perpendicular to i + j + k is

• $\frac{1}{\sqrt{2}}\left(\mathrm{j} + \mathrm{k}\right)$

• $\frac{1}{\sqrt{3}}\left(\mathrm{i} - \mathrm{j} + \mathrm{k}\right)$

• $\frac{1}{\sqrt{2}}\left(\mathrm{j} - \mathrm{k}\right)$

• $\frac{1}{\sqrt{3}}\left(\mathrm{i} + \mathrm{j} - \mathrm{k}\right)$

If a and b are two non-zero perpendicular vectors, then a vector y satisfying equations a . y = c (where, c is scalar) and a x y = b is

• ${\left|\mathrm{a}\right|}^2\left[\mathrm{ca} - \left(\mathrm{a} \times \mathrm{b}\right)\right]$

• ${\left|\mathrm{a}\right|}^2\left[\mathrm{ca} + \left(\mathrm{a} \times \mathrm{b}\right)\right]$

• $\frac{1}{{\left|\mathrm{a}\right|}^2}\left[\mathrm{ca} - \left(\mathrm{a} \times \mathrm{b}\right)\right]$

• $\frac{1}{{\left|\mathrm{a}\right|}^2}\left[\mathrm{ca} + \left(\mathrm{a} \times \mathrm{b}\right)\right]$

Three non-zero non-collinear vectors $\hat{\mathrm{a}}, \hat{\mathrm{b}} \mathrm{and} \hat{\mathrm{c}}$ are such that $\hat{\mathrm{a}} + 3\hat{\mathrm{b}}$ is collinear with $\hat{\mathrm{c}}$, while $\hat{\mathrm{c}}$ is $3\hat{\mathrm{b}} + 2\hat{\mathrm{c}}$ collinear with a. Then $\hat{\mathrm{a}} + 3\hat{\mathrm{b}} + 2\hat{\mathrm{c}}$ equals

• 0

• $2\hat{\mathrm{a}}$

• $3\hat{\mathrm{b}}$

• $4\hat{\mathrm{c}}$

If $\hat{\mathrm{a}}, \hat{\mathrm{b}} \mathrm{and} \hat{\mathrm{c}}$ are non-coplanar vectors and if $\hat{\mathrm{d}}$ is such that $\hat{\mathrm{d}} = \frac{1}{\mathrm{x}}\left(\hat{\mathrm{a}} + \hat{\mathrm{b}} + \hat{\mathrm{c}}\right)$ and $\hat{\mathrm{d}} = \frac{1}{\mathrm{y}}\left(\hat{\mathrm{b}} + \hat{\mathrm{c}} + \hat{\mathrm{d}}\right)$ where x and y are non-zero real numbers, then $\frac{1}{\mathrm{xy}}\left(\hat{\mathrm{a}} + \hat{\mathrm{b}} + \hat{\mathrm{c}} + \hat{\mathrm{d}}\right)$ equals to

• 3c

• - a

• 0

• 2a

If a, b and c are vectors with magnitudes 2, 3 and 4 respectively, then the best upper bound of ${\left|\hat{\mathrm{a}} - \hat{\mathrm{b}}\right|}^2 + {\left|\hat{\mathrm{b}} - \hat{\mathrm{c}}\right|}^2 + {\left|\hat{\mathrm{c}} - \hat{\mathrm{a}}\right|}^2$ among the given values is

• 93

• 97

• 87

• 90

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

• 3

• 10

• 9

• 6

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

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

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

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

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

470.

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

• 2

• 3

• 4

• 5