Calculus
Homework # 8
Homework # 8
9.5 # 44: (4 pts, p. 672)
(a) Find parametric equations for the line of intersection of the planes
(b) Find the angle between the planes
3x − 2y + z = 1
2x + y − 3z = 3
See Example 7 on p. 668 for this. Because the domain of these planes includes all values of
z, find the point on the line of intersection of the planes where z = 0 by setting z = 0 and then
solving the resulting two linear equations for x and y. (Note any of the three variables x, y, or
z could be used for this and that any value for z or the selected variable would work for this,
but 0 is the easiest to work with). Setting z = 0 gives:
3x − 2y = 1
2x + y = 3
1
3
x = 3
2
3
2
y = 3
2
−2
1
7
= =1
7
−2
1
1
3
7
= =1
7
−2
1
Cramer’s Rule
From this, the point (1, 1, 0) is on the line of intersection. Now the line of intersection must be
perpendicular to the normals for both planes. These normals are (from the equations of the
planes):
n1 = ⟨3, −2, 1⟩
n2 = ⟨2, 1, −3⟩
To find a vector which is perpendicular to both of these normals, take the cross product of
these two normals:
i j
k n1 × n2 = 3 −2 1 = (6 − 1)i + (2 + 9)j + (3 + 4)k = 5i + 11j + 7k
2 1 −3
Using the point (1, 1, 0) which is on the line and the vector 5i + 11j + 7k which is parallel to
the line gives this equation for the line and a corresponding parametric equation for the line
using t as the parameter (see p. 663).
y−1
z
x−1
=
=
5
11
7
⟨x, y, z⟩ = ⟨1 + 5t, 1 + 11t, 7t⟩
Equation for the line
Parametric equation for the line
1
Calculus
Homework # 8
The angle between the two planes is the same as the angle between the normals of the two
planes. Those normals are (from the equations of the planes):
n1 = ⟨3, −2, 1⟩
n2 = ⟨2, 1, −3⟩
Use the dot product to find the angle between these two normals:
n1 · n2
|n1 | |n2 |
6−2−3
√
=√
9+4+1 4+1+9
1
1
=√ √
=
14
14 14
1
θ = cos−1 14
≈ 1.4993 radians ≈ 85.9040◦
cos θ =
9.6 # 23: (3 pts, p. 681)
(a) What does the equation x2 + y 2 = 1 represent as a curve in R2 ?
(b) What does it represent as a curve in R3 ?
(c) What does the equation x2 + z 2 = 1 represent?
Part (a): In R2 , this equation represents a circle centered at the origin with a radius of 1.
Part (b): In R3 , this equation represents a vertical cylinder centered at the origin (or the
z-axis) with a radius of 1.
Part (c): In R3 , this equation represents a cylinder centered at the y-axis with a radius of
1.
9.7 # 22: (3 pts, p. 687) Write the equation (a) in cylindrical coordinates and (b) in spherical
coordinates.
x2 + y 2 + z 2 = 2
Part (a): The equations for converting to cylindrical coordinates are:
r = x2 + y 2
y
θ = tan−1
x
z=z
x = r cos θ
y = r sin θ
z=z
No conversion
Substituting these into the equation gives:
x2 + y 2 + z 2 = 2
2
2
2
2
Original equation
2
Substituting
2
Simplifying
r cos θ + r sin θ + z = 2
2
r +z =2
2
Calculus
Homework # 8
Part (b): The equations for converting to cylindrical coordinates are:
ρ = x2 + y 2 + z 2
y
θ = tan−1
x
z
ϕ = cos−1 √
x2 + y 2 + z 2
x = ρ sin ϕ cos θ
y = ρ sin ϕ sin θ
z = ρ cos ϕ
Substituting these into the equation gives:
x2 + y 2 + z 2 = 2
ρ2 sin2 ϕ cos2 θ + ρ2 sin2 ϕ sin2 θ + ρ2 cos2 ϕ = 2
2
2
2
2
ρ sin ϕ + ρ cos ϕ = 2
2
ρ =2
3
Original equation
Substituting
Simplifying