# Intersection of a Line and a Plane

Here you will learn how to find intersection of a line and a plane with examples.

Let’s begin –

## Intersection of a Line and a Plane

Let the equation of a line be $$x – x_1\over l$$ = $$y – y_1\over m$$ = $$z – z_1\over n$$ and that of a plane be ax + by + cz + d = 0.

The coordinates of any point on the line  $$x – x_1\over l$$ = $$y – y_1\over m$$ = $$z – z_1\over n$$ is given

$$x – x_1\over l$$ = $$y – y_1\over m$$ = $$z – z_1\over n$$ = r (say)

or, $$(x_1 + lr, y_1 + mr, z + nr)$$                …………(i)

If it lies on the plane ax + by + cz + d = 0, then

$$a(x_1 + lr)$$ + $$b(y_1 + mr)$$ + $$c(z_1 + nr)$$ + d = 0

$$\implies$$ $$(ax_1 + by_1 + cz_1 + d)$$ + r(al + bm + cn) = 0

$$\implies$$ r = -$$(ax_1 + by_1 + cz_1 + d)\over al + bm + cn$$

Substituting the value of r in (i), we obtain the coordinates of the required point of intersection.

In order to find the coordinates of the point of intersection of a line and a plane, we may use the following algorithm,

Algorithm :

1). Write the coordinates of any point on the line in terms of some parameters r (say).

2). Substitute these coordinates in the equation of the plane to obtain the value of r.

3). Put the value of r in the coordinates of the point in step 1.

Example : Find the coordinates of the point where the line through the points A(3, 4, 1) and B(5, 1, 6) crosses the XY-plane.

Solution : The equation of the line passing through A and B is

$$x – 3\over 5 – 3$$ = $$y – 4\over 1 – 4$$ = $$z – 1\over 6 – 1$$

or,  $$x – 3\over 2$$ = $$y – 4\over -3$$ = $$z – 1\over 5$$

The coordinates of any point on this line are given by

$$x – 3\over 2$$ = $$y – 4\over -3$$ = $$z – 1\over 5$$ = $$\lambda$$

$$\implies$$ x = $$2\lambda + 3$$,  y = $$-3\lambda + 4$$,  z = $$5\lambda + 1$$

So, $$(2\lambda + 3, -3\lambda + 4, 5\lambda + 1)$$ are coordinates of any point on the line passing through A and B. If it lies on XY-plane i.e z = 0.Then.

$$5\lambda + 1$$ = 0 $$\implies$$ $$\lambda$$ = -$$1\over 5$$

Thus, the coordinates of the required point are (13/5, 23/5, 0).