List of top Mathematics Questions on Three Dimensional Geometry asked in CBSE Class XII

Find a point on the line \( \frac{x - 2}{3} = \frac{1 - y}{2} = \frac{z - 3}{2} \) at a distance of \( \sqrt{2} \) units from the point \( (1, 2, 3) \).

Find the value of $p$ if the shortest distance between the lines \[ \vec{r} = (\hat{i} + 2\hat{j} + \hat{k}) + \lambda(\hat{i} - \hat{j} + \hat{k}) \quad \text{and} \quad \vec{r} = (p\hat{i} - \hat{j} - \hat{k}) + \mu(2\hat{i} + \hat{j} + 2\hat{k}) \] is exactly $\frac{3}{\sqrt{2}}$ units.

Find the coordinates of the foot of the perpendicular drawn from the point $(0, 2, 3)$ onto the spatial line given by: \[ \frac{-x-3}{-5} = \frac{1-y}{-2} = \frac{3z+12}{9} \] And hence, find the exact geometric length of this perpendicular line segment.

Find the exact angular measurement $\theta$ between the following pair of spatial lines in three-dimensional space: \[ \frac{x-2}{3} = \frac{y+5}{2} = \frac{1-z}{-6} \quad \text{and} \quad \frac{x-7}{1} = \frac{y}{2} = \frac{6-z}{-2} \]