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Determine intersections of lines and planes

Using coordinate, vector, and parametric equations in three dimensions, the learner determines the common points of lines and planes by solving and interpreting the resulting linear equations. Consistency distinguishes a unique intersection, no intersection, and infinitely many common points; for two lines, this includes intersecting, parallel, and skew cases, while two planes may intersect in a line, be parallel, or coincide. The focus is on linear objects in three-dimensional space, not nonlinear intersections or higher-dimensional generalizations.

Detailed Explanation: 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

r=(120)+t(2−11)\mathbf r= \begin{pmatrix} 1\\ 2\\ 0 \end{pmatrix} +t \begin{pmatrix} 2\\ -1\\ 1 \end{pmatrix}

and the plane

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

The line can be written in coordinate form as

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

Substitute these expressions into the plane equation:

(1+2t)+(2−t)+t=5.(1+2t)+(2-t)+t=5.

Simplify:

3+2t=5.3+2t=5.

Therefore,

2t=2⟹t=1.2t=2 \qquad\Longrightarrow\qquad t=1.

Now substitute t=1t=1 into the line:

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

So the line and plane have the unique intersection point

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

A single value of tt gives one intersection point. If substitution produces a contradiction, such as 3=53=5, there is no intersection. If the equation is always true, such as 3=33=3, every point on the line lies in the plane, so there are infinitely many common points.

Learn by doing: Determine intersections of lines and planes

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    3D Space - Plane and Line - Scalar and Vector Equations to Intersection Point

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