Skill: Determine intersections of lines and planes

To find the intersection of a line and a plane:

1. Write the coordinates of a general point on the line using its parameter.
2. Substitute those coordinates into the plane equation.
3. Solve for the parameter.
4. Substitute the parameter back into the line equation to find the intersection point.
5. Check whether the result gives one point, no points, or infinitely many points.

Example

Find the intersection of the line

$$\mathbf r= \begin{pmatrix} 1\\ 2\\ 0 \end{pmatrix} +t \begin{pmatrix} 2\\ -1\\ 1 \end{pmatrix}$$

and the plane

$$x+y+z=5.$$

The line can be written in coordinate form as

$$x=1+2t,\qquad y=2-t,\qquad z=t.$$

Substitute these expressions into the plane equation:

$$(1+2t)+(2-t)+t=5.$$

Simplify:

$$3+2t=5.$$

Therefore,

$$2t=2 \qquad\Longrightarrow\qquad t=1.$$

Now substitute $t=1$ into the line:

$$x=1+2(1)=3,$$ $$y=2-1=1,$$ $$z=1.$$

So the line and plane have the unique intersection point

$$\boxed{(3,1,1)}.$$

A single value of $t$ gives one intersection point. If substitution produces a contradiction, such as $3=5$, there is no intersection. If the equation is always true, such as $3=3$, every point on the line lies in the plane, so there are infinitely many common points.