Line-Plane Intersection — Definition, Formula & Examples

A line-plane intersection is the point where a line in three-dimensional space passes through a plane. If the line is not parallel to the plane, it crosses the plane at exactly one point.

Given a line defined parametrically as $\mathbf{r}(t) = \mathbf{p} + t\mathbf{d}$ and a plane defined by $\mathbf{n} \cdot (\mathbf{r} - \mathbf{q}) = 0$ , the intersection occurs at the parameter value $t = \frac{\mathbf{n} \cdot (\mathbf{q} - \mathbf{p})}{\mathbf{n} \cdot \mathbf{d}}$ , provided $\mathbf{n} \cdot \mathbf{d} \neq 0$ . If $\mathbf{n} \cdot \mathbf{d} = 0$ , the line is parallel to the plane and either misses it entirely or lies within it.

Key Formula

t = \frac{\mathbf{n} \cdot (\mathbf{q} - \mathbf{p})}{\mathbf{n} \cdot \mathbf{d}}

Where:

$t$ = Parameter value at which the line meets the plane
$\mathbf{n}$ = Normal vector of the plane
$\mathbf{p}$ = A known point on the line
$\mathbf{d}$ = Direction vector of the line
$\mathbf{q}$ = A known point on the plane

How It Works

To find the intersection, express the line in parametric form using a point and a direction vector. Substitute the parametric equations into the equation of the plane. Solve the resulting equation for the parameter

t

. Plug

t

back into the parametric equations to get the coordinates of the intersection point. If the denominator

\mathbf{n} \cdot \mathbf{d}

equals zero, the line is parallel to the plane and no unique intersection exists.

Worked Example

Problem: Find where the line through (1, 0, 2) with direction vector (0, 1, -1) intersects the plane 2x + y + z = 10.

Write parametric equations: Express the line parametrically using the given point and direction.

x = 1,\quad y = t,\quad z = 2 - t

Substitute into the plane equation: Replace x, y, and z in 2x + y + z = 10 with the parametric expressions.

2(1) + t + (2 - t) = 10 \implies 4 = 10

Interpret the result: The equation 4 = 10 is a contradiction, which means the line is parallel to the plane and never intersects it. Check: the normal vector is (2, 1, 1) and the direction is (0, 1, -1). Their dot product is 0 + 1 - 1 = 0, confirming the line is parallel.

\mathbf{n} \cdot \mathbf{d} = 2(0) + 1(1) + 1(-1) = 0

Answer: No intersection exists — the line is parallel to the plane and does not lie in it.

Why It Matters

Ray-plane intersection is fundamental in computer graphics, where every pixel on screen is determined by casting a ray and finding where it strikes a surface. It also appears in multivariable calculus and linear algebra when analyzing geometric relationships in three dimensions.

Common Mistakes

Mistake: Forgetting to check whether the line is parallel to the plane before solving for t.

Correction: Always compute the dot product of the plane's normal vector and the line's direction vector first. If it equals zero, no unique intersection point exists.