If a, b and c are three non-zero vectors such that each one of them being perpendicular to the sum of the other two vectors, then the value of ${\left|\mathrm{a} + \mathrm{b} + \mathrm{c}\right|}^2$ is

• ${\left|\mathrm{a}\right|}^2 + {\left|\mathrm{b}\right|}^2 + {\left|\mathrm{c}\right|}^2$

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

• 2$\left({\left|\mathrm{a}\right|}^2 + {\left|\mathrm{b}\right|}^2 + {\left|\mathrm{c}\right|}^2\right)$

• $\frac{1}{2}\left({\left|\mathrm{a}\right|}^2 + {\left|\mathrm{b}\right|}^2 + {\left|\mathrm{c}\right|}^2\right)$

Let u, v and w be vectors such that u + v + w = 0. If $\left|\mathrm{u}\right| = 3, \left|\mathrm{v}\right| = 4 \mathrm{and} \left|\mathrm{w}\right| = 5$, then u - v + v · w + w · u is equal to

• 0

• - 25

• 25

• 50

If $\mathrm{\lambda }\left(3\hat{\mathrm{i}} + 2\hat{\mathrm{j}} - 6\hat{\mathrm{k}}\right)$ is a unit vector, then the value of $\mathrm{\lambda }$ are

• $\pm \frac{1}{7}$

• $\pm 7$

• $\pm \sqrt{43}$

• $\pm \frac{1}{\sqrt{43}}$

If the direction cosines of a vector of magnitude 3 are $\frac{2}{3}, \frac{- \mathrm{a}}{3}, \frac{2}{3}, \mathrm{a} >\hspace{0.17em}0$, then the vector is

• $2\hat{\mathrm{i}} + \hat{\mathrm{j}} + 2\hat{\mathrm{k}}$

• $2\hat{\mathrm{i}} - \hat{\mathrm{j}} + 2\hat{\mathrm{k}}$

   

• $\hat{\mathrm{i}} - 2\hat{\mathrm{j}} + 2\hat{\mathrm{k}}$

• $\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 2\hat{\mathrm{k}}$

465.

Equation of the plane through the mid-point of the line segment joining the points P(4, 5, - 10), Q(- 1, 2, 1) and perpendicular to PQ is

• $\mathrm{r} . \left(\frac{3}{2}\hat{\mathrm{i}} + \frac{7}{2}\hat{\mathrm{j}} - \frac{9}{2}\hat{\mathrm{k}}\right) = 45$

• $\mathrm{r} . \left(- \hat{\mathrm{i}} + 2\hat{\mathrm{j}} + \hat{\mathrm{k}}\right) = \frac{135}{2}$

• $\mathrm{r} . \left(5\hat{\mathrm{i}} +\hspace{0.17em}3\hat{\mathrm{j}} - 11\hat{\mathrm{k}}\right) + \frac{135}{2} = 0$

• $\mathrm{r} . \left(5\hat{\mathrm{i}} + 3\hat{\mathrm{j}} - 11\hat{\mathrm{k}}\right) = \frac{135}{2}$

A unit vector parallel to the straight line $\frac{\mathrm{x} - 2}{3} = \frac{3 + \mathrm{y}}{- 1} = \frac{\mathrm{z} - 2}{- 4}$ is

• $\frac{1}{\sqrt{26}}\left(3\hat{\mathrm{i}} - \hat{\mathrm{j}} + 4\hat{\mathrm{k}}\right)$

• $\frac{1}{\sqrt{26}}\left(\hat{\mathrm{i}} + 3\hat{\mathrm{j}} - \hat{\mathrm{k}}\right)$

• $\frac{1}{\sqrt{26}}\left(3\hat{\mathrm{i}} - \hat{\mathrm{j}} - 4\hat{\mathrm{k}}\right)$

• $\frac{1}{\sqrt{26}}\left(3\hat{\mathrm{i}} + \hat{\mathrm{j}} + 4\hat{\mathrm{k}}\right)$

The angle between the two vectors $\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}$ and $2\hat{\mathrm{i}} - 2\hat{\mathrm{j}} + 2\hat{\mathrm{k}}$ is equal to

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

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

• ${\cos }^{-1}\left(\frac{5}{6}\right)$

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

If a = $\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}$, b = $4\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + 4\hat{\mathrm{k}}$ and $\mathrm{c} = \hat{\mathrm{i}} + \mathrm{\alpha }\hat{\mathrm{j}} + \mathrm{\beta }\hat{\mathrm{k}}$ are copalnar and $\left|\mathrm{c}\right| = \sqrt{3}$, then

• $\mathrm{\alpha } = \sqrt{2}, \mathrm{\beta } = 1$

• $\mathrm{\alpha } = 1, \mathrm{\beta } = \pm 1$

• $\mathrm{\alpha } = \pm 1, \mathrm{\beta } = 1$

• $\mathrm{\alpha } = \pm 1, \mathrm{\beta } = - 1$

Let P (1, 2, 3) and Q (- 1, - 2, - 3) be the two points and let O be the origin. Then, $\left|\mathrm{PQ} + \mathrm{OP}\right|$ is equal to

• $\sqrt{13}$

• $\sqrt{14}$

• $\sqrt{24}$

• $\sqrt{12}$

Let ABCD be a parallelogram. If $\mathrm{AB} = \hat{\mathrm{i}} + 3\hat{\mathrm{j}} + 7\hat{\mathrm{k}}$, AD = $2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} - 5\hat{\mathrm{k}}$ and p is a unit vector parallel to AC, then p is equal to

• $\frac{1}{3}\left(2\hat{\mathrm{i}} + \hat{\mathrm{j}} + 2\hat{\mathrm{k}}\right)$

• $\frac{1}{3}\left(2\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 2\hat{\mathrm{k}}\right)$

• $\frac{1}{7}\left(3\hat{\mathrm{i}} + 6\hat{\mathrm{j}} + 2\hat{\mathrm{k}}\right)$

• $\frac{1}{7}\left(6 + 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}}\right)$