If d = a x b + b x c + c x a is a non-zero vector and $\left|(\mathrm{d} · \mathrm{c}) (\mathrm{a} \mathrm{x} \mathrm{b}) + (\mathrm{d} · \mathrm{a}) (\mathrm{b} \mathrm{x} \mathrm{c}) + (\mathrm{d} . \mathrm{b}) (\mathrm{c} \mathrm{x} \mathrm{a})\right|$ = 0, then

• $\left|\mathrm{a}\right| + \left|\mathrm{b}\right| + \left|\mathrm{c}\right| = \left|\mathrm{d}\right|$

• $\left|\mathrm{a}\right| = \left|\mathrm{b}\right| = \left|\mathrm{c}\right|$

• a, b and c are coplanar

• None of the above

Let u, v and w be such that $\left|\mathrm{u}\right|$ = 1, $\left|\mathrm{v}\right|$ = 2 and $\left|\mathrm{w}\right|$ = 3. If the projection of v along u is equal to that of w along u and vectors v and w are perpendicular to each other, then $\left|\mathrm{u} - \mathrm{v} + \mathrm{w}\right|$ equals

• 2

• $\sqrt{7}$

• $\sqrt{14}$

• 14

If a and b are two vectors, such that a . b < 0 and $\left|\mathrm{a} . \mathrm{b}\right| = \left|\mathrm{a} \times \mathrm{b}\right|$ then the angle between vectors a and b is

• $\frac{3\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{4}$

• $\mathrm{\pi }$

• $\frac{7\mathrm{\pi }}{4}$

A tetrahedron has vertices at 0(0, 0, 0), A(1, - 2, 1), B (-2, 1, 1) and C (1, - 1, 2). Then, the angle between the faces OAB and ABC will be

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

• ${\cos }^{-1}\left(\frac{- 1}{6}\right)$

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

• ${\cos }^{-1}\left(\frac{1}{4}\right)$

If a, b and c are three non-coplanar vectors, then (a + b - c) . [(a - b) x (b - c)] equals

• 0

• a . b $\times$ c

• a . c $\times$ b

• 3a . b $\times$ c

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

398.

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