Algebraic Form
In vector notation, a plane can be expressed as the set of points for which
where is a normal vector to the plane and is a point on the plane. The vector equation for a line is
where is a vector in the direction of the line, is a point on the line, and is an scalar in the real number domain. Substitute the line into the plane equation to get
Distribute to get
And solve for
If the line starts outside the plane and is parallel to the plane, there is no intersection. In this case, the above denominator will be zero and the numerator will be non-zero. If the line starts inside the plane and is parallel to the plane, the line intersects the plane everywhere. In this case, both the numerator and denominator above will be zero. In all other cases, the line intersects the plane once and represents the intersection as the distance along the line from, i.e.
Read more about this topic: Line-plane Intersection
Famous quotes related to algebraic form:
“I have no scheme about it,—no designs on men at all; and, if I had, my mode would be to tempt them with the fruit, and not with the manure. To what end do I lead a simple life at all, pray? That I may teach others to simplify their lives?—and so all our lives be simplified merely, like an algebraic formula? Or not, rather, that I may make use of the ground I have cleared, to live more worthily and profitably?”
—Henry David Thoreau (1817–1862)