1.Find the direction cosines of a line which makes equal angles with the positive coordinate axes.
2.Find the direction cosines of the line passing through the two points $(-2, 4, -5)$ and $(1, 2, 3)$.
3.If a line has direction ratios $2, -1, -2$, determine its direction cosines.
4.Show that the points $A(2, 3, -4), B(1, -2, 3)$ and $C(3, 8, -11)$ are collinear using direction ratios.
5.A line makes angles $90^\circ, 135^\circ, 45^\circ$ with the x, y, and z axes respectively. Find its direction cosines.
6.Find the direction cosines of the line joining the points $(1, 0, 0)$ and $(0, 1, 1)$.
7.Find the vector equation of the line passing through the point $(5, 2, -4)$ and parallel to the vector $3\hat{i} + 2\hat{j} - 8\hat{k}$.
8.Find the Cartesian equation of the line passing through the point $(-2, 4, -5)$ and parallel to the line given by $\frac{x+3}{3} = \frac{y-4}{5} = \frac{z+8}{6}$.
9.The Cartesian equation of a line is $\frac{x-5}{3} = \frac{y+4}{7} = \frac{z-6}{2}$. Write its vector equation.
10.Find the equation of the line in vector and Cartesian form that passes through the origin and $(5, -2, 3)$.
11.Find the points on the line $\frac{x-1}{2} = \frac{y+2}{3} = \frac{z-3}{6}$ at a distance of $7$ units from the point $(1, -2, 3)$.
12.Find the Cartesian equation of a line passing through $(1, 2, 3)$ and parallel to the x-axis.
13.Find the angle between the lines $\vec{r} = 2\hat{i} - 5\hat{j} + \hat{k} + \lambda(3\hat{i} + 2\hat{j} + 6\hat{k})$ and $\vec{r} = 7\hat{i} - 6\hat{k} + \mu(\hat{i} + 2\hat{j} + 2\hat{k})$.
14.Find the angle between the lines $\frac{x}{2} = \frac{y}{2} = \frac{z}{1}$ and $\frac{x-5}{4} = \frac{y-2}{1} = \frac{z-3}{8}$.
15.Find the value of $p$ so that the lines $\frac{1-x}{3} = \frac{7y-14}{2p} = \frac{z-3}{2}$ and $\frac{7-7x}{3p} = \frac{y-5}{1} = \frac{6-z}{5}$ are at right angles.
16.Show that the lines $\frac{x-5}{7} = \frac{y+2}{-5} = \frac{z}{1}$ and $\frac{x}{1} = \frac{y}{2} = \frac{z}{3}$ are perpendicular to each other.
17.Find the angle between the pair of lines whose direction ratios are $(1, 1, 2)$ and $(\sqrt{3}-1, -\sqrt{3}-1, 4)$.
18.Find the value of $\lambda$ if the lines $\frac{x-1}{-3} = \frac{y-2}{2\lambda} = \frac{z-3}{2}$ and $\frac{x-1}{3\lambda} = \frac{y-1}{1} = \frac{z-6}{-5}$ are perpendicular.
19.Find the shortest distance between the lines $\vec{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(\hat{i} - 3\hat{j} + 2\hat{k})$ and $\vec{r} = (4\hat{i} + 5\hat{j} + 6\hat{k}) + \mu(2\hat{i} + 3\hat{j} + \hat{k})$.
20.Find the shortest distance between the lines given by $\frac{x+1}{7} = \frac{y+1}{-6} = \frac{z+1}{1}$ and $\frac{x-3}{1} = \frac{y-5}{-2} = \frac{z-7}{1}$.
21.Find the shortest distance between the parallel lines $\vec{r} = (\hat{i} + \hat{j}) + \lambda(2\hat{i} - \hat{j} + \hat{k})$ and $\vec{r} = (2\hat{i} + \hat{j} - \hat{k}) + \mu(2\hat{i} - \hat{j} + \hat{k})$.
22.Show that the lines $\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$ and $\frac{x-4}{5} = \frac{y-1}{2} = z$ intersect.
23.Find the shortest distance between the lines passing through points $A(1, 2, 3), B(2, 4, 5)$ and $C(-1, 0, 1), D(0, 1, 2)$.
24.Find the coordinates of the foot of the perpendicular drawn from the point $P(0, 2, 3)$ to the line $\frac{x+3}{5} = \frac{y-1}{2} = \frac{z+4}{3}$.
25.Find the perpendicular distance from the point $(1, 0, 0)$ to the line $\frac{x-1}{2} = \frac{y+1}{-3} = \frac{z+10}{8}$.
26.Find the image of the point $(1, 6, 3)$ in the line $\frac{x}{1} = \frac{y-1}{2} = \frac{z-2}{3}$.
27.Find the coordinates of the foot of the perpendicular drawn from the point $(2, 4, -1)$ to the line $\frac{x+5}{1} = \frac{y+3}{4} = \frac{z-6}{-9}$.
28.Find the length of the perpendicular drawn from the origin to the line $\frac{x+2}{2} = \frac{y-3}{-2} = \frac{z}{-1}$.
29.Find the vector equation of a plane which is at a distance of $7$ units from the origin and normal to the vector $3\hat{i} + 5\hat{j} - 6\hat{k}$.
30.Find the Cartesian equation of the plane passing through the point $(2, 3, 4)$ and perpendicular to the vector $3\hat{i} - 2\hat{j} + 6\hat{k}$.
31.Find the equation of the plane passing through the non-collinear points $(1, 1, 0), (1, 2, 1),$ and $(-2, 2, -1)$.
32.Find the intercepts cut off by the plane $2x - 3y + 4z = 12$ on the coordinate axes.
33.Find the equation of the plane with intercepts $2, 3,$ and $-4$ on the $x, y,$ and $z$ axes respectively.
34.Reduce the equation of the plane $x - 2y + 2z = 6$ to normal form, and hence find the perpendicular distance from the origin.
35.Find the equation of the plane through $(1, 0, -2)$ and parallel to the plane $2x + 3y - z = 5$.
36.Find the angle between the planes $2x + y - 2z = 5$ and $3x - 6y - 2z = 7$.
37.Find the angle between the line $\frac{x+1}{2} = \frac{y}{3} = \frac{z-3}{6}$ and the plane $10x + 2y - 11z = 3$.
38.Find the coordinates of the point of intersection of the line $\frac{x-2}{3} = \frac{y+1}{4} = \frac{z-2}{1}$ and the plane $x - y + z = 5$.
39.Find the coordinates of the point where the line passing through $(3, -4, -5)$ and $(2, -3, 1)$ crosses the plane $2x + y + z = 7$.
40.Find the value of $k$ if the plane $x - 2y + kz = 4$ is parallel to the line $\frac{x}{2} = \frac{y-1}{3} = \frac{z+1}{4}$.
41.Show that the lines $\vec{r} = (\hat{i} + \hat{j} - \hat{k}) + \lambda(3\hat{i} - \hat{j}) $ and $\vec{r} = (4\hat{i} - \hat{k}) + \mu(2\hat{i} + 3\hat{k})$ are coplanar.
42.Find the distance of the point $(2, 5, -3)$ from the plane $6x - 3y + 2z - 4 = 0$.
43.Find the distance between the parallel planes $2x - y + 3z + 4 = 0$ and $6x - 3y + 9z - 3 = 0$.
44.Find the distance from the origin to the plane $3x - 4y + 12z = 26$.
45.Find the perpendicular distance of the point $(1, 2, 3)$ from the plane $x + 2y - 3z + 5 = 0$.
