NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry
EXERCISE 11.1
- If a line makes angles 90°, 135°, 45° with the x, y and z-axes respectively, find its direction cosines.
- Find the direction cosines of a line which makes equal angles with the coordinate axes.
- If a line has the direction ratios –18, 12, – 4, then what are its direction cosines ?
- Show that the points (2, 3, 4), (– 1, – 2, 1), (5, 8, 7) are collinear.
- Find the direction cosines of the sides of the triangle whose vertices are (3, 5, – 4), (– 1, 1, 2) and (– 5, – 5, – 2).
EXERCISE 11.2
- Show that the three lines with direction cosines
\(\frac{12}{13}, \frac{-3}{13}, \frac{-4}{13}, \frac{4}{13}, \frac{12}{13}, \frac{3}{13}, \frac{3}{13}, \frac{-4}{13}, \frac{12}{13}\) are mutually perpendicular.
- Show that the line through the points (1, – 1, 2), (3, 4, – 2) is perpendicular to the line through the points (0, 3, 2) and (3, 5, 6).
- Show that the line through the points (4, 7, 8), (2, 3, 4) is parallel to the line through the points (– 1, – 2, 1), (1, 2, 5).
- Find the equation of the line which passes through the point (1, 2, 3) and is parallel to the vector \(3\hat{i} + 2\hat{j} – 2\hat{k}\) .
- Find the equation of the line in vector and in cartesian form that passes through the point with position vector \(2\hat{i} – \hat{j} + 4\hat{k}\) and is in the direction \(\hat{i} + 2\hat{j} – \hat{k}\).
- Find the cartesian equation of the line which passes through the point (– 2, 4, – 5) and parallel to the line given by \(\frac{x + 3}{3} = \frac{y – 4}{5} = \frac{z + 8}{6}\).
