Mathematics : Three Dimensional Geometry

The equation of the plane passing through (- 1, 5, - 7) and parallel to the plane 2x - 5y + 7z + 11 = 0, is

• $\mathrm{r}. \left(2\hat{\mathrm{i}} - 5\hat{\mathrm{j}} - 7\hat{\mathrm{k}}\right) + 76 = 0$

• $\mathrm{r}. \left(2\hat{\mathrm{i}} - 5\hat{\mathrm{j}} + 7\hat{\mathrm{k}}\right) + 76 = 0$

• $\mathrm{r}. \left(2\hat{\mathrm{i}} - 5\hat{\mathrm{j}} -+ 7\hat{\mathrm{k}}\right) + 75 = 0$

• $\mathrm{r}. \left(2\hat{\mathrm{i}} - 5\hat{\mathrm{j}} + 7\hat{\mathrm{k}}\right) + 65 = 0$

The angle subtended at the point (1, 2, 3) by the points P(2, 4, 5) and Q(3, 3, 1) is

• 90°

• 60°

• 30°

• 0°

If the two lines $\frac{\mathrm{x} - 1}{2} = \frac{1 - \mathrm{y}}{- \mathrm{a}} = \frac{\mathrm{z}}{4}$ and $\frac{\mathrm{x} - 3}{1} = \frac{2\mathrm{y} - 3}{4} = \frac{\mathrm{z} - 2}{2}$ are perpendicular, then the value of a is equal to

• - 4

• 5

• - 5

• 4

If the line $\frac{\mathrm{x} + 1}{2} = \frac{\mathrm{y} + 1}{3} = \frac{\mathrm{z} + 1}{4}$ meets the plane x + 2y + 3z = 14 at P, then the distance between P and the origin is

• $\sqrt{14}$

• $\sqrt{15}$

• $\sqrt{13}$

• $\sqrt{12}$

The point of intersection of the straight lines $\mathrm{r} = \left(3\hat{\mathrm{i}} - 4\hat{\mathrm{j}} + 5\hat{\mathrm{k}}\right) + \mathrm{\lambda }\left(- \hat{\mathrm{i}} - 2\hat{\mathrm{j}} + 2\hat{\mathrm{k}}\right)$ and $\frac{3 - \mathrm{x}}{- 1} = \frac{\mathrm{y} + 4}{2} = \frac{\mathrm{z} - 5}{7}$ is

• (- 3, - 4, - 5)

• (- 3, 4, 5)

• (- 3, 4, - 5)

• (3, - 4, 5)

The vector equation of the straight line $\frac{\mathrm{x} - 2}{- 1} = \frac{\mathrm{y}}{- 3} = \frac{1 - \mathrm{z}}{2}$ is

• $\mathrm{r} = \left(2\hat{\mathrm{i}} + \hat{\mathrm{k}}\right) + \mathrm{t}\left(\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + 2\hat{\mathrm{k}}\right)$

• $\mathrm{r} = \left(2\hat{\mathrm{i}} - \hat{\mathrm{k}}\right) + \mathrm{t}\left(\hat{\mathrm{i}} - 3\hat{\mathrm{j}} - 2\hat{\mathrm{k}}\right)$

• $\mathrm{r} = \left(2\hat{\mathrm{i}} + \hat{\mathrm{k}}\right) + \mathrm{t}\left(\hat{\mathrm{i}} - 3\hat{\mathrm{j}} + 2\hat{\mathrm{k}}\right)$

• $\mathrm{r} = \left(2\hat{\mathrm{i}} + \hat{\mathrm{k}}\right) + \mathrm{t}\left(\hat{\mathrm{i}} - 3\hat{\mathrm{j}} - 2\hat{\mathrm{k}}\right)$

The straight line $\mathrm{r} = (\hat{\mathrm{i}} + \hat{\mathrm{j}} + 2\hat{\mathrm{k}}) + \mathrm{t}\left(2\hat{\mathrm{i}} + 5\hat{\mathrm{j}} + 3\hat{\mathrm{k}}\right)$ is parallel to the plane $\mathrm{r} . (2\hat{\mathrm{i}} + \hat{\mathrm{j}} - 3\hat{\mathrm{k}}) = 5$. Then, the distance between the straight line and the plane is

• $\frac{9}{\sqrt{14}}$

• $\frac{8}{\sqrt{14}}$

• $\frac{7}{\sqrt{14}}$

• $\frac{6}{\sqrt{14}}$

The distance between (2, 1, 0) and 2x + y + Zz + 5 = 0 is

• 10

• 10/3

• 10/9

• 5

The equation of the plane that passes through the points (1, 0, 2), (-1, 1, 2), (5, 0, 3) is

• x + 2y - 4z + 7 = 0

• x + 2y - 3z + 7 = 0

• x - 2y + 4z + 7 = 0

• 2y - 4z - 7 + x = 0

If a, b, c are vectors such that a + b + c = 0 and $\left|\mathrm{a}\right| = 7, \left|\mathrm{b}\right| = 5, \left|\mathrm{c}\right| = 3$, then the angle between c and b is

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right.$

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$

• $\mathrm{\pi }$