| Lines, Planes, and Vectors |
Lines, Planes, and Vectors
In this tutorial, we will use vector methods to represent lines and planes in
3-space.
A D V E R T I S E M E N T
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)
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
Example
The line through (2,-1,3) and parallel to the vector v = (3,-7,4) has
parametric equations
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 |
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Thus, the graph of the equation
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 |
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Simplifying,
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
- 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
- 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 |
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