Calc 2 Lecture Notes
Section 10.5
Page 1 of 10
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:
Given a line l containing a given point P1 = (x1, y1, z1) and an arbitrary point P = (x, y, z) on the
line, and a vector a  a1 , a2 , a3 that is parallel to l:
PP
 ta
1
x  x1 , y  y1 , z  z1  ta1 , ta2 , ta3
 x  x1  ta1
y  y1  ta2
z  z1  ta3
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
Calc 2 Lecture Notes
Section 10.5
Page 2 of 10
Practice:
1. Find parametric equations for the line through the point (3, 1, 4) and parallel to the vector
a  2,7,1 .
2. Find parametric equations for the line containing the points (1, 1, 2) and (5, -2, 7).
Calc 2 Lecture Notes
Section 10.5
Page 3 of 10
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
Page 4 of 10
Equation for a plane in 3D:
Given a plane p containing a given point P1 = (x1, y1, z1) and an arbitrary point P = (x, y, z) in the
plane, and a vector a  a1 , a2 , a3 that is normal to p:
PP
1 a  0
a1 , a2 , a3  x  x1 , y  y1 , z  z1  0
 a1  x  x1   a2  y  y1   a3  z  z1   0
Linear 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 if you divide through by a1, that the equation only has three parameters, which can be
solved for as an alternative way of finding the equation of a plane given three points.
a1 x  a2 y  a3 z  d  0
x
a
a2
d
y 3 z 0
a1
a1
a1
x  ay  bz  c  0
Calc 2 Lecture Notes
Section 10.5
Page 5 of 10
Practice:
4. Find the equation of the plane containing the point (3, 1, 4) and perpendicular to the
vector 2,7,1 , then sketch the plane.
Calc 2 Lecture Notes
Section 10.5
Page 6 of 10
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
Page 7 of 10
Intersection of planes:
Two unique planes (in three dimensions) are either parallel 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,
equate one of the variables of the two planes, then write a second variable in terms of the third
variable, which will be the independent parameter for the parametric form.
Practice:
6. Find the equation of the plane containing the point (3, 5, 4) and parallel to the plane
2x + 7y – z = 5.
Calc 2 Lecture Notes
Section 10.5
Page 8 of 10
7. Find the intersection of the planes 2x + 7y – z = 5 and 3x – 5y + z = 4.
Calc 2 Lecture Notes
Section 10.5
Page 9 of 10
Distance between a point and a plane:
Let P1 = (x0, y0, z0) be a point in the plane described by ax + by + cz + d = 0, and P0 = (x1, y1, z1)
be a point off the plane. Then a = a, b, c is a normal vector to the plane, and the distance from
the point to the plane is compa P1 P0 
P1 P0  a
.
a
Practice:
8. What is the distance between the point (3, 1, 4) and the plane x + y + z = 4?
9. What is the distance between the parallel planes 2x – 3y + z = 6 and 4x – 6y + 2z = -8?
Calc 2 Lecture Notes
Section 10.5
Page 10 of 10