Lines, Planes, and Vectors

In this tutorial, we will use vector methods to represent lines and planes in 3-space.

Displacement Vector

The displacement vector v with initial point (x1,y1,z1) and terminal point (x2,y2,z2) is

v = (x2-x1,y2-y1,z2-z1)

Why?

That is, if vector v were positioned with its initial point at the origin, then its terminal point would be at (x2-x1,y2-y1,z2-z1).

Example

The vector v with initial point (-1,4,5) and final point (4,-3,2) is

v = ( 4-(-1),-3-4,2-5 ) = (5,-7,-3)

Parametric Equations for a Line in 3-space

The line through the point (x0,y0,z0) and parallel to the non-zero vector v = (a,b,c) has parametric equations

x
=
x0 + at
y
=
y0 + bt
z
=
z0 + ct

Why?

Example

The line through (2,-1,3) and parallel to the vector v = (3,-7,4) has parametric equations

x
=
2+3t
y
=
-1-7t
z
=
3+4t

Notice that when t = 0, we are at the point (2,-1,3). As t increases or decreases from 0, we move away from this point parallel to the direction indicated by (3,-7,4).

If you know two points p1 = (x1,y1,z1) and p2 = (x2,y2,z2) that a line passes through, you can find a parameterization for the line. First, find the displacement vector v = (x2-x1,y2-y1,z2-z1). then write down parametric equations for the line through either p1 or p2 and parallel to v.

Equation of a Plane in 3-space

The equation of the plane containing the point (x0,y0,z0) with normal vector n = (a,b,c) is

a(x-x0)+ b(y-y0)+c(z-z0) = 0

Why?

Thus, the graph of the equation

ax+by+cz = d
is a plane with normal vector (a,b,c).

Example

The equation of the plane containing (2,4,-1) and normal to the vector n = (3,5,-2) is

3(x-2)+5(y-4)-2(z-(-1)) = 0
Simplifying,
3x+5y-2z = 28.

With a little extra work, we can use this procedure to find the equation of the plane defined by any three points. First, compute displacement vectors u and v between two pairs of these points. Then n = u × v is normal to the plane. Now, use one of the points and the vector n = u × v to obtain the equation of the plane.

Key Concepts [index]

  • Displacement Vector

    The displacement vector v with initial point (x1,y1,z1) and terminal point (x2,y2,z2) is v = (x2-x1,y2-y1,z2-z1).

  • Parametric Equations for a line in 3-space

    The line through the point (x0,y0,z0) and parallel to the non-zero vector v = (a,b,c) has parametric equations

    x
    =
    x0 + at
    y
    =
    y0 + bt
    z
    =
    z0 + ct

  • Equation of a plane in 3-space

    The equation of the plane containing the point (x0,y0,z0) with normal vector n = (a,b,c) is

    a(x-x0)+ b(y-y0)+c(z-z0) = 0