6.2.2 Combinations of Lines & Planes

Intersection of a Line and a Plane

Let the equation of the line be:

$$\mathbf{r} = \mathbf{a} + \lambda \mathbf{b}$$

and the equation of the plane be:

$$\mathbf{r} \times \mathbf{n} = d$$

where:

• $\mathbf{a}$ is a point on the line.
• $\mathbf{b}$ is the direction vector of the line.
• $\mathbf{n}$ is the normal vector to the plane.

Finding the Intersection:

Substitute the line equation into the plane equation**:**

$$(\mathbf{a} + \lambda \mathbf{b}) \times \mathbf{n} = d$$

Solve for $\lambda$:

$$\lambda = \frac{d - (\mathbf{a} \times \mathbf{n})}{\mathbf{b} \times \mathbf{n}}$$

Find the intersection point**:**

Substitute $\lambda$ back into the line equation.

Perpendicular Distance from a Point to a Plane

Let the plane equation be:

$$ax + by + cz + d = 0$$

and the point be $(x_1, y_1, z_1)$

Perpendicular Distance Formula:

$$d = \frac{|ax_1 + by_1 + cz_1 + d|}{\sqrt{a^2 + b^2 + c^2}}$$

Worked Example