MATH 251: Calculus 3, SET8

12: Vectors and the Geometry of Space

12.1: Three-Dimensional Coordinate Systems

These problems are done with the CAS. See Hand Solutions for details.

1. [796/6]

The equation in represents a plane parallel to , the xz-plane.
The equation in represents a plane parallel to , the xy-plane.
The pair of equations in is the intersection of these two planes, a line parallel to the x-axis ().
syms s3 s5 x y z
s3 = sym(3); s5 = sym(5);
figure
fsurf(x, 3, z, [-5 5 -5 10], 'MeshDensity', 7, 'FaceColor', 'g')
grid on; hold on
fsurf(x, y, 5, 'MeshDensity', 7, 'FaceColor', 'r')
fplot3(x, s3, s5, [-7 7], 'm', 'LineWidth', 7)
axis equal; axis([-7 7 -5 5 0 10])
xticks(-5:5:5); yticks([-5 0 3 5]); zticks(0:5:10)
xlabel('x'); ylabel('y'); zlabel('z')
title('SET8 796/6')

2. [796/8]

The equation represents a circular cylinder of radius with axis of symmetry the y-axis. The surface is of infinite extent in the y-direction.
syms y theta
x = 3*cos(theta), z = 3*sin(theta)
x = 
z = 
figure
fsurf(x, y, z, [0 2*pi -5 5], 'MeshDensity', 12)
axis equal; axis([-5 5 -5 5 -5 5])
xticks(-3:3:3); yticks(-5:5:5); zticks(-3:3:3)
xlabel('x'); ylabel('y'); zlabel('z')
title('SET8 796/8')

3. [797/14]

The sphere with center and radius has equation .
syms phi theta
x = 2 + 5*sin(phi)*cos(theta), y = -6 + 5*sin(phi)*sin(theta), z = 4 + 5*cos(phi)
x = 
y = 
z = 
figure
fsurf(x, y, z, [0 pi 0 2*pi], 'MeshDensity', 20); hold on
x = 2 + 3*cos(theta), y = -6 + 3*sin(theta)
x = 
y = 
fplot3(x, y, sym(0), [0 2*pi], 'r', 'LineWidth', 4)
s21 = sym(sqrt(21)), y = -6 + s21*cos(theta), z = 4 + s21*sin(theta)
s21 = 
y = 
z = 
fplot3(sym(0), y, z, [0 2*pi], 'm', 'LineWidth', 4)
axis equal; alpha 0.2
xlabel('x'); ylabel('y'); zlabel('z')
title('SET8 796/14')

4. [797/16]

The sphere with center that passes through the origin has radius . Its equation is.
syms phi theta
s14 = sym(sqrt(14))
s14 = 
x = 1 + s14*sin(phi)*cos(theta), y = 2 + s14*sin(phi)*sin(theta), z = 3 + s14*cos(phi)
x = 
y = 
z = 
figure
fsurf(x, y, z, [0 pi 0 2*pi], 'm', 'MeshDensity', 20)
axis equal
xlabel('x'); ylabel('y'); zlabel('z')
title('SET8 796/16')

5. [797/20]

Subtract the right side from the left side of the equation and set it equal to zero. Divide by 3, complete squares, and factor to obtain the equation of a sphere in standard form, . Its center is and its radius is .
syms x y z
eq1 = 3*x^2 + 3*y^2 + 3*z^2 == 10 + 6*y + 12*z
eq1 = 
eq2 = lhs(eq1) - rhs(eq1) == 0
eq2 = 
eq3 = eq2/3
eq3 = 
eq4 = eq3 + 1 + 4 + 10/3
eq4 = 
eq = (x-0)^2 + (y-1)^2 + (z-2)^2 == sqrt(25/3)^2
eq = 
%
syms phi rho theta
rho = sym(sqrt(25/3))
rho = 
x = 0 + rho*sin(phi)*cos(theta), y = 1 + rho*sin(phi)*sin(theta), z = 2 + rho*cos(phi)
x = 
y = 
z = 
figure
fsurf(x, y, z, [0 pi 0 2*pi], 'c', 'MeshDensity', 20)
axis equal
xlabel('x'); ylabel('y'); zlabel('z')
title('SET8 797/20')

6. [797/34]

The inequality represents all points in space on and inside the radius 2 sphere with center at the origin.
syms phi theta
x = 2*sin(phi)*cos(theta), y = 2*sin(phi)*sin(theta), z = 2*cos(phi)
x = 
y = 
z = 
figure
fsurf(x, y, z, [0 pi 0 2*pi], 'r', 'MeshDensity', 20)
axis equal; alpha 0.3
xticks(-2:2:2); yticks(-2:2:2); zticks(-2:2:2)
xlabel('x'); ylabel('y'); zlabel('z')
title('SET8 797/34')

7. [797/38]

The inequality or represents all points in space outside the sphere of radius 1 with center .

8. [797/42]

The solid upper hemisphere of radius 2 centered at the origin satisfies the inequalities .
syms r phi theta
figure
x = r*cos(theta), y = r*sin(theta), z = sym(0)
x = 
y = 
z = 0
fsurf(x, y, z, [0 2 0 2*pi], 'r', 'MeshDensity', 20); hold on
axis equal
x = 2*sin(phi)*cos(theta), y = 2*sin(phi)*sin(theta), z = 2*cos(phi)
x = 
y = 
z = 
fsurf(x, y, z, [0 pi/2 0 2*pi], 'y', 'MeshDensity', 20)
alpha 0.4
xlabel('x'); ylabel('y'); zlabel('z')
title('SET8 797/42')

9. [797/44]

Points whose distance from is twice the distance from P to satisfy . We'll square the corresponding resulting equation then complete squares to obtain , which represents a sphere with center and radius .
syms x y z
A = sym([-1 5 3]), B = sym([6 2 -2]), P = [x y z]
A = 
B = 
P = 
eq1 = norm(P-A)^2 == (2*norm(P-B))^2
eq1 = 
eq2 = (x + 1)^2 - 4*(x - 6)^2 - 4*(y - 2)^2 + (y - 5)^2 - 4*(z + 2)^2 + (z - 3)^2 == 0
eq2 = 
eq3 = expand(eq2)
eq3 = 
eq4 = eq3/(-3)
eq4 = 
eq5 = eq4 + (25/3)^2 + 1^2 + (11/3)^2 - 47
eq5 = 
eq = (x-25/3)^2 + (y-1)^2 + (z-(-11/3))^2 == sqrt(25/3)^2
eq = 
%
syms phi rho theta
rho = sym(sqrt(332/9))
rho = 
x = 25/3 + rho*sin(phi)*cos(theta), y = 1 + rho*sin(phi)*sin(theta), z = -11/3 + rho*cos(phi)
x =