Mathematics : Three Dimensional Geometry

The ratio in which yz-plane divides the line segment joining ( - 3, 4, - 2) and (2, 1, 3) is

• - 4 : 1

• 3 : 2

• - 2 : 3

• 1 :4

The cosine of the angle A of the tnangle with vertices

A(1, - 1, 2), B(6, 11, 2), C(1, 2, 6) is

• $\raisebox{1ex}{$63$}\!\left/ \!\raisebox{-1ex}{$65$}\right.$

• $\raisebox{1ex}{$36$}\!\left/ \!\raisebox{-1ex}{$65$}\right.$

• $\raisebox{1ex}{${\displaystyle 16}$}\!\left/ \!\raisebox{-1ex}{${\displaystyle 65}$}\right.$

• $\raisebox{1ex}{$13$}\!\left/ \!\raisebox{-1ex}{$64$}\right.$

The angle between the lines whose direction 

cosine are $\left(\frac{\sqrt{3}}{4}, \frac{1}{4}, \frac{\sqrt{3}}{2}\right) \mathrm{and} \left(\frac{\sqrt{3}}{4}, \frac{1}{4}, \frac{- \sqrt{3}}{2}\right), \mathrm{is}$

• $\mathrm{\pi }$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{4}$

In a quadrilateral ABCD, the point P divides DC in the ratio 1 : 2 and Q is the mid point of AC. If$\overrightarrow{\mathrm{AB}} + 2\overrightarrow{\mathrm{AD}} + \overrightarrow{\mathrm{BC}} - 2\overrightarrow{\mathrm{DC}} = \mathrm{k}\overrightarrow{\mathrm{PQ}},$ then k is equal

• - 6

• - 4

• 6

• 4

The angle between the lines whose direction cosines satisfy the equations l + m + n = 0, l₂ + m₂ - n₂ = 0 is

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

$$\begin{gathered} \mathrm{If} \overrightarrow{\mathrm{a}} = - \hat{\mathrm{i}} + \hat{\mathrm{j}} + 2\hat{\mathrm{k}}, \overrightarrow{\mathrm{b}} = 2\hat{\mathrm{i}} - \hat{\mathrm{j}} - \hat{\mathrm{k}}, \mathrm{and} \overrightarrow{\mathrm{c}} = - 2\hat{\mathrm{i}} + \hat{\mathrm{j}} + 3\hat{\mathrm{k}}, \\ \mathrm{then} \mathrm{the} \mathrm{angle} \mathrm{between} 2\overrightarrow{\mathrm{a}} - \overrightarrow{\mathrm{c}} \mathrm{and} \overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} \mathrm{is} \end{gathered}$$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{3\mathrm{\pi }}{2}$

$$\begin{gathered} \mathrm{Suppose} \overrightarrow{\mathrm{a}} = \mathrm{\lambda }\hat{\mathrm{i}} - 7\hat{\mathrm{j}} + 3\hat{\mathrm{k}}, \overrightarrow{\mathrm{b}} = \mathrm{\lambda }\hat{\mathrm{i}} + \hat{\mathrm{j}} + 2\mathrm{\lambda }\hat{\mathrm{k}}. \mathrm{If} \\ \mathrm{the} \mathrm{angle} \mathrm{between} \mathrm{a} \mathrm{and} \mathrm{b} \mathrm{is} \mathrm{greater} \mathrm{than} 90°, \mathrm{then} \mathrm{A} \mathrm{satisfies} \mathrm{the} \mathrm{inequality} \end{gathered}$$

• $- 7 < \mathrm{\lambda } < 1$

• $\mathrm{\lambda } > 1$

• $1 < \mathrm{\lambda } < 7$

• $- 5 < \mathrm{\lambda } < 1$

$$\begin{gathered} \mathrm{The} \mathrm{volume} \mathrm{of} \mathrm{the} \mathrm{tetrahedron} \mathrm{having} \mathrm{the} \mathrm{edge} \\ \hat{\mathrm{i}} + 2\hat{\mathrm{j}} - \hat{\mathrm{k}}, \hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}, \hat{\mathrm{i}} - \hat{\mathrm{j}} + \mathrm{\lambda }\hat{\mathrm{k}} \mathrm{as} \mathrm{coterminous}, \mathrm{is} \frac{2}{3}\mathrm{cubic} \mathrm{unit}. \mathrm{Then} \mathrm{\lambda } = ? \end{gathered}$$

• 1

• 2

• 3

• 4

The image of the point (3, 2, 1) in the plane 2x - y + 3z = 7 is

• (1, 2, 3)

• (2, 3, 1)

• (3, 2, 1)

• (2, 1, 3)

A point moves in the xy-plane such that the sum of its distance from two mutually perpendicular lines is always equal to 5 units. The area(in sq units) enclosed by the locus of the point,is

• $\frac{25}{4}$

• 25

• 50

• 100