If the dot product of the line’s direction vector with the plane’s normal is zero, the direction is parallel to the plane. To tell whether the line lies in the plane or sits outside it, substitute one point on the line into the plane equation.
When the direction is parallel to the plane, a point that satisfies the equation means the whole line lies in the plane. A point that fails the equation means the line stays outside the plane and has no intersection with it.
- Read the plane’s normal vector from the coefficients of x, y and z.
- Take its dot product with the line’s direction vector.
- If the result is zero, test any one point on the line.
Worked example | A parallel line outside the plane
Let the plane be x + 2y − z = 5 and the line be (x, y, z) = (1, 1, 0) + t(2, −1, 0), with real t. The normal is (1, 2, −1). Its dot product with (2, −1, 0) is 2 − 2 = 0.
Substitute the base point (1, 1, 0). The plane’s left side gives 1 + 2 − 0 = 3. Since it must equal 5, the line is parallel to the plane with no intersection. Every point on the line gives the same left-side value because the dot product is zero.
Keep the direction and change the base point
Now use (3, 1, 0) + t(2, −1, 0). The direction is unchanged, so the dot product is still zero. The new base point gives 3 + 2 = 5. The whole line now lies in the plane.
Direct substitution shows the same result. Along the second line, the left side is (3 + 2t) + 2(1 − t) = 5 for every real t.
What if the dot product is nonzero?
For the same plane, use the line (1, 1, 0) + t(1, 0, 0). Find its intersection with the plane.
Answer. Substitute to get 3 + t = 5, so t = 2. The unique intersection is (3, 1, 0).
Frequently asked questions
Is checking a point enough on its own?
It tells you whether that point lies in the plane. Check the direction too to decide whether the line stays in the plane or crosses it.
Why does a nonzero dot product give one intersection?
Substitution gives a linear equation in t with a nonzero coefficient. It has exactly one solution.
Official information
SEAB, 2027 GCE A-Level H2 Mathematics syllabus (9758), Three-Dimensional Vector Geometry, relationships between a line and a plane. Syllabus checked 11 November 2026. The examples are original.