Calc 2 Lecture Notes
Section 10.5
Page 1 of 8
Section 10.5: Lines and Planes in Space
Big idea: Vectors can be used to easily derive the equation for a line or plane in three
dimensions.
Big skill: You should be able to derive the equation for a line or plane in 3D given a variety of
information, and calculate line-line, line-plane, and plane-plane intersections.
Parametric equations for a line in 3D:
To find the parametric equations for the line l that that contains a given point P1 = (x1, y1, z1) and
that is parallel to a given vector a  a1 , a2 , a3 , use the fact that the displacement vector between
the given point P1 and any arbitrary point P = (x, y, z) on the line must be parallel to a:
PP
 ta
1
x  x1 , y  y1 , z  z1  ta1 , ta2 , ta3
 x  x1  ta1
y  y1  ta2
z  z1  ta3
A line l that is parallel to the vector a and that
contains the point P1.
Symmetric equations for a line in 3D:
x  x1
y  y1
z  z1
t
t
t
a1
a2
a3

x  x1 y  y1 z  z1


a1
a2
a3
Practice:
1. Find equations of the line through the point (3, 1, 4) and parallel to the vector 2,7,1 .
Calc 2 Lecture Notes
Section 10.5
Page 2 of 8
2. Find equations of the line containing the points (1, 1, 2) and (5, -2, 7).
3. Determine whether the previous two lines intersect.
Definition 5.1: Relationships Between lines and their Parallel Vectors
Let l1 and l2 be two lines in 3 , with parallel vectors a and b, respectively, and let  be the angle
between a and b.
(i). The lines l1 and l2 are parallel whenever a and b are parallel.
(ii). If lines l1 and l2 intersect, then
a. The angle between l1 and l2 is , and
b. The lines l1 and l2 are orthogonal whenever a and b are orthogonal.
Definition 5.2: Skew Lines
Nonparallel, nonintersecting lines are called skew lines.
Calc 2 Lecture Notes
Section 10.5
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Equation for a plane in 3D:
To find the equation for the plane p that that contains a given point P1 = (x1, y1, z1) and that is
perpendicular to a given vector a  a1 , a2 , a3 , use the fact that the displacement vector between
the given point P1 and any arbitrary point P = (x, y, z) in the plane must be perpendicular to a,
which means their dot product must be zero:
PP
1 a  0
a1 , a2 , a3  x  x1 , y  y1 , z  z1  0
 a1  x  x1   a2  y  y1   a3  z  z1   0
A plane p that is perpendicular to the vector a,
and that contains the point P1.
Alternative Equation for a plane in 3D:
a1  x  x1   a2  y  y1   a3  z  z1   0
a1 x  a2 y  a3 z   a1 x1  a2 y1  a3 z1   0
constant
a1 x  a2 y  a3 z  d  0
Note that the coefficients of the variables are the corresponding components of the perpendicular
vector. Thus, we can find the components of the normal vector to a plane whose equation is
given just by reading off the coefficients.
Also note that if you divide the above equation through by a1, then the equation has only three
parameters.
a1 x  a2 y  a3 z  d  0
x
a
a2
d
y 3 z 0
a1
a1
a1
x  ay  bz  c  0
If you are given three coplanar points, then you can solve for these three parameters (a, b, and c)
using standard algebraic techniques instead of using vector algebra.
Calc 2 Lecture Notes
Section 10.5
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Practice:
4. Find the equation of the plane containing the point (3, 1, 4) and that is perpendicular to
the vector 2,7,1 , then sketch the plane.
5. Find the equation of the plane containing the points (3, 1, 4), (2, 7, 1), (-1, -2, -3).
Calc 2 Lecture Notes
Section 10.5
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Calc 2 Lecture Notes
Section 10.5
Page 6 of 8
Intersection of planes:
Two planes (in three dimensions) are either parallel (this includes when they completely
overlap), or they intersect in a straight line. The angle made by intersecting planes is the same as
the angle between their normal vectors. Specifically, planes are parallel when their normal
vectors are parallel, and planes are orthogonal when their normal vectors are orthogonal.
To find the line (expressed in parametric form) that is the intersection of two planes,
eliminate one variable from the two equations, then solve the remaining equation for one of its
variables in terms of the other variable. Let the “other variable” be the independent parameter,
then write the remaining variables in terms of the independent parameter.
Practice:
6. Find the equation of the plane containing the point (3, 1, 4) and parallel to the plane
2x + y + z = 1.
7. Find the intersection of the planes 2x + y + z = 1 and 3x – 5y + z = 4.
Calc 2 Lecture Notes
Section 10.5
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Distance between a point and a plane:
To find the distance between a given point P0 = (x0, y0, z0) and a plane p that that contains a point
P1 = (x1, y1, z1) and that has a normal vector a, compute the projection of the displacement vector
from P1 to P0 onto the vector a:
If the plane has equation
ax  by  cz  d  0 ,
then a normal vector is:
a  a, b, c .
The displacement vector from P1 to P0 is:
PP
1 0  x0  x1 , y0  y1 , z0  z1 ,
so the distance between the point and plane is:
d  compa P1 P0
d
P1 P0  a
a
Practice:
8. What is the distance between the point (3, 1, 4) and the plane x + y + z = 4?
Calc 2 Lecture Notes
Section 10.5
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9. What is the distance between the parallel planes 2x – 3y + z = 6 and 4x – 6y + 2z = 8?