 # Three Dimensional Geometry Class 12 Notes - Chapter 11

## Direction Cosine of Line

This is referred to the cosine of the angles subtended by a line on the positive direction of the coordinate axis. If we are given a line whose direction cosines are p,q,r, then $p^{2}+q^{2}+r^{2}=1$ and suppose we have line joining two points such as R($x_{1},y_{1},z_{1}$) and S($x_{2},y_{2},z_{2}$) are represented by

$x_{2}-x_{1}/RS,y_{2}-y_{1}/RS,z_{2}-z_{1}/RS$ and RS=$\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}$

If a line has p,q,r are the direction cosines and a,b,c are the direction ratios then, p=$a/\sqrt{a^{2}+b^{2}+c^{2}}$, q=$b/\sqrt{a^{2}+b^{2}+c^{2}}$,r=$c/\sqrt{a^{2}+b^{2}+c^{2}}$

### What are Skew Lines?

These are referred to the lines in space which neither intersecting nor parallel. They exist in a different plane. The angle between two lines which intersect each other can be found from any point which is parallel to each of the skew lines.

If $p_{1},q_{1},r_{1}$ and $p_{2},q_{2},r_{2}$ are the given direction cosines and angle between two distinct lines is $\theta$ then,

$cos\theta$=$\left | p_{1}p_{2}+q_{1}q_{2}+r_{1}r_{2} \right |$

And if two lines have direction ratios as $l_{1},m_{1},n_{1}$ and $l_{2},m_{2},n_{2}$ and angle $\theta$ is given then,

$\left | l_{1}l_{2}+m_{1}m_{2}+n_{1}n_{2} \right/\sqrt{l_{1}^{2}+m_{1}^{2}+n_{1}^{2} \sqrt{l_{2}^{2}+m_{2}^{2}+n_{2}^{2}}}$

### The equation of the vector of a line

The equation of vector of the line passing through a point having position vector as $\underset{a}{\rightarrow}$ and is parallel to a given vector $\underset{b}{\rightarrow}$ is $\underset{r}{\rightarrow}$=$\underset{a}{\rightarrow}+\lambda \underset{b}{\rightarrow}$Equation of a plane which is at a distance of d from the origin and the direction
cosines of the normal to the plane as l, m, n is lx + my + nz = d and Shortest distance between two skew lines is the line segment perpendicular to both the lines.

### Cartesian Equation of a Line passing through two points

If there are two points with dimensions $(x_{1},y_{1},z_{1}) and (x_{2},y_{2},z_{2})$  then $x-x_{1}/x_{2}-x_{1}=y-y_{1}/y_{2}-y_{1}=z-z_{1}/z_{2}-z_{1}$

### Important Questions

1. Distance between the two planes with dimensions 2x + 3y + 4z = 4 and 4x + 6y + 8z = 12 is
2. (A) 2 units (B) 4 units (C) 8 units (D) $2/\sqrt{29}$ units

1. If P be the origin and the coordinates of P be (2, 3, – 4), then find the equation of the plane passing through Q and perpendicular to PQ.
2. Find the coordinates of the point where the line through (4, – 5, – 6) and(3, – 4, 2) crosses the plane 3x + y + z = 8.
3. Find the coordinates of the point where the line through (6, 2, 7) and (4, 5,2) crosses the YZ-plane
4. Find the angle between the lines whose direction ratios are a, b, c and b – c, c – a, a – b.