Chapter 12 – Vectors and the
Geometry of Space
12.5 Equations of Lines and Planes
Objectives:
 Find vector, parametric,
and general forms of
equations of lines and
planes.
 Find distances and angles
between lines and planes
12.5 Equations of Lines and Planes
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Lines in 2D

A line in the xy-plane is determined
when a point on the line and the
direction of the line (its slope or
angle of inclination) are given.
12.5 Equations of Lines and Planes
2
Lines in 3D
A
line L in 3D space is determined
when we know:
◦ A point P0(x0, y0, z0) on L
◦ The direction of L, given by a vector
12.5 Equations of Lines and Planes
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Definition – Vector Equation
So, we let v be a vector parallel to L.



Let P(x, y, z) be an arbitrary point on L.
Let r0 and r be the position vectors of P0 and P.
If a is the vector with representation from P to Po.
Then the Triangle Law for vector addition gives r = r0 + a.
However, since a and v are parallel vectors, there is a scalar t
such that a = tv.
So we have
1
r = r0 + t v
This is a vector equation of L.
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Definition continued
We can also write this definition in component form:
r = <x, y, z> and r0 = <x0, y0, z0>
◦ So, vector Equation 1 becomes:
2
<x, y, z> = <x0 + ta, y0 + tb, z0 + tc>
Positive values of t correspond to
points on L that lie on one side of P0.
Negative values
correspond to points that
lie on the other side.
12.5 Equations of Lines and Planes
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Definition continued

Two vectors are equal if and only if corresponding
components are equal.

Hence, we have the following three scalar equations.
x = x0 + at y = y0 + bt z = z0 + ct, where t ℝ

These equations are called parametric equations of the line
L through the point P0(x0, y0, z0) and parallel to the vector
v = <a, b, c>.

Each value of the parameter t gives a point (x, y, z) on L.
12.5 Equations of Lines and Planes
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Example 1 – pg 824 #2

Find a vector equation and
parametric equations for the line.
The line through the point (6,-5,2)
and parallel to the vector <1,3,-2/3>.
12.5 Equations of Lines and Planes
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Definition - Symmetric Equations

If we solve the equations for t
x = x0 + at, y = y0 + bt, z = z0 + ct,
we get the following symmetric equations.
3
12.5 Equations of Lines and Planes
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Example 2

Find parametric equations and
symmetric equations for the line.
The line through the points (6,1,-3)
and (2,4,5).
12.5 Equations of Lines and Planes
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Equations of Line Segments

The line segment from r0 to r1 is
given by the vector equation
4
r(t) = (1 – t)r0 + t r1
where 0 ≤ t ≤ 1
12.5 Equations of Lines and Planes
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Definition – Skew Lines

Lines that are skew
are not parallel and
do NOT intersect.
They do NOT lie in
the same plane.
12.5 Equations of Lines and Planes
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Example 3

Determine whether the lines L1 and
L2 are parallel, skew, or intersecting.
If they intersect, find the point of
intersection.
x 1 y  3 z  2
L1 :


2
2
1
x2 y6 z 2
L2 :


1
1
3
12.5 Equations of Lines and Planes
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Planes

A plane in space is determined by:
◦ A point P0(x0, y0, z0) in the plane
◦ A vector n that is orthogonal to the plane
12.5 Equations of Lines and Planes
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Definition – Normal Vector
This orthogonal vector n is called
a normal vector.
 The normal vector n is orthogonal to
every vector in the given plane.
 In particular, n is orthogonal to r – r0.

12.5 Equations of Lines and Planes
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Equation of Planes

If n is orthogonal to r – r0 we have the
following equations:

Either Equation 5 or Equation 6
is called a vector equation of the
plane.
12.5 Equations of Lines and Planes
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Equations of Planes

This equation is the scalar equation of the
plane through P0(x0, y0, z0) with normal
vector n = <a, b, c>.
12.5 Equations of Lines and Planes
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Linear Equation
where d = –(ax0 + by0 + cz0)
◦ This is called a linear equation
in x, y, and z.
12.5 Equations of Lines and Planes
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Example 4 – pg. 825 # 34

Find an equation of the plane.
The plane that passes through the
point (1,2,3) and contains the line
x = 3t, y = 1 + t, z = 2 – t.
12.5 Equations of Lines and Planes
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Example 5 – pg. 825 # 45

Find the point at which the line
intersects the given plane.
x=3–t
y=2+t
z = 5t
x – y + 2z = 9
12.5 Equations of Lines and Planes
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Example 6 – pg. 825 #48

Where does the line through (1,0,1)
and (4, -2, 2) intersect the plane
x + y + z = 6?
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More Examples
The video examples below are from
section 12.5 in your textbook. Please
watch them on your own time for
extra instruction. Each video is
about 2 minutes in length.
◦ Example 3
◦ Example 4
◦ Example 7
12.5 Equations of Lines and Planes
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Demonstrations
Feel free to explore these
demonstrations below.

Constructing Vector Geometry
Solutions
12.5 Equations of Lines and Planes
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