For any three vectors a, b and c [a + b, b + c, c + a] is

• [a, b, c]

• 3[a, b, c]

• 2[a, b, c]

• 0

If $\mathrm{r} = \mathrm{\alpha b} \times \mathrm{c} + \mathrm{\beta c} \times \mathrm{a} + \mathrm{\gamma a} \times \mathrm{b}$ and [a b c] = 2, then $\mathrm{\alpha } + \mathrm{\beta } + \mathrm{\gamma }$ is equal to

• $\mathrm{r} . \left[\mathrm{b} \times \mathrm{c} + \mathrm{c} \times \mathrm{a} + \mathrm{a} \times \mathrm{b}\right]$

• $\frac{1}{2}\mathrm{r} . \left(\mathrm{a} +\hspace{0.17em}\mathrm{b} + \mathrm{c}\right)$

• 2r . (a + b + c)

• 4

If a, b, c are three non-coplanar vectors and p, q, rare reciprocal vectors, then (la + mb + nc) · (lp + mq + nr) is equal to

• l + m + n

• l³ + m³ + n³

• l² + m² + n²

• None of these

The vector b = 3j + 4k is to be written as the sum of a vector b₁ parallel to a = i + j and a vector b₂ perpendicular to a. Then, b₁ is equal to

• $\frac{3}{2}$(i + j)

• $\frac{2}{3}$(i + J)

• $\frac{1}{2}$(i + j)

• $\frac{1}{3}$(i + j)

If a = i + j + k, b = i + 3j + 5k and c = 7i + 9j + 11k, then the area of parallelogram having diagonals a + b and  b + c is

• 4$\sqrt{6}$ sq units

• $\frac{1}{2}\sqrt{21}$ sq units

• $\frac{\sqrt{6}}{2}$ sq units

• $\sqrt{6}$ sq units

56.

The projection of the vector i - 2j + k on the vector 4i - 4j + 7k is

• $\frac{5\sqrt{6}}{10}$

• $\frac{19}{9}$

• $\frac{9}{19}$

• $\frac{\sqrt{6}}{19}$

If a, b, c are three non-zero vectors such that a + b+ c = 0 and m = a . b + b . c + c · a, then

• m < 0

• m > 0

• m = 0

• m = 3

If $\overrightarrow{\mathrm{a}} = 2\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$, $\overrightarrow{\mathrm{b}} = - \hat{\mathrm{i}} + 2\hat{\mathrm{j}} + \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{c}} = 3\hat{\mathrm{i}} + \hat{\mathrm{j}}$ then $\overrightarrow{\mathrm{a}} + \mathrm{t}\overrightarrow{\mathrm{b}}$ is perpendicular to c, if t is equal to

• 2

• 4

• 6

• 8

The distance between the line $\overrightarrow{\mathrm{r}} = 2\hat{\mathrm{i}} - 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}} + \mathrm{\lambda }\left(\hat{\mathrm{i}} - \hat{\mathrm{j}} + 4\hat{\mathrm{k}}\right)$ and the plane $\overrightarrow{\mathrm{r}} . \left( \hat{\mathrm{i}} + 5\hat{\mathrm{j}} + \hat{\mathrm{k}}\right) = 5$, is

• $\frac{10}{3}$

• $\frac{10}{\sqrt{3}}$

• $\frac{10}{3\sqrt{3}}$

• $\frac{10}{9}$

If m₁, m₂, m₃ and m₄ are respectively the magnitudes of the vectors

$$\begin{gathered} \overrightarrow{{\mathrm{a}}_1} = 2\hat{\mathrm{i}} - \hat{\mathrm{j}} + \hat{\mathrm{k}}, \overrightarrow{{\mathrm{a}}_2} = 3\hat{\mathrm{i}} - 4\hat{\mathrm{j}} - 4\hat{\mathrm{k}}, \\ \overrightarrow{{\mathrm{a}}_3} = \hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}} \mathrm{and} \overrightarrow{{\mathrm{a}}_4} = - \hat{\mathrm{i}} + 3\hat{\mathrm{j}} + \hat{\mathrm{k}}, \end{gathered}$$

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