MATH 251: Calculus 3, SET8

13: Vectors Functions

13.1: Vector Functions and Space Curves

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

1. [853/4]

The limit command renders the needful.
syms t
r = [(t^2 -t) / (t-1) sqrt(t+8), sin(pi*t) / log(t)]
r = 
L = limit(r, t, 1)
L = 

2. [854/10]

The space curve is a circular helix with axis of symmetry the y-axis. As t increases, the curve progresses in the direction of the positive y-axis.
syms t
x = sin(pi*t), y = t, z = cos(pi*t)
x = 
y = t
z = 
figure
fplot3(x,y,z, [0 6], 'LineWidth', 2)
axis equal; xlabel('x'); ylabel('y'); zlabel('z')
title('SET8, 854/10')

3. [854/22]

Another space curve is plotted.
syms t
x = cos(t), y = sin(t), z = 1/(1+t^2)
x = 
y = 
z = 
figure
fplot3(x,y,z, [-4*pi 4*pi], 'r', 'LineWidth', 2)
xlabel('x'); ylabel('y'); zlabel('z')
title('SET8, 854/22')

4. [854/28]

The space curve is the curve of intersection of a circular cylinder with a parabolic cylinder.
syms t x y z R T X Y Z
x = sin(t), y = cos(t), z = sin(t)^2
x = 
y = 
z = 
surf1 = z == x^2, surf2 = x^2 + y^2 == 1 % Substitution yields true statments.
surf1 = 
surf2 = 
% The space curve lies on both surfaces. It is their curve of intersection.
figure
fsurf(X, Y, X^2, [-1 1 -1 1], 'y', 'MeshDensity', 9); hold on
fsurf(cos(T), sin(T), Z, [0 2*pi -1 1], 'c', 'MeshDensity', 13)
fplot3(x,y,z, [0 4*pi], 'm', 'LineWidth', 5)
alpha 0.3
axis equal; xlabel('x'); ylabel('y'); zlabel('z')
xticks(-1:1); yticks(-1:1); zticks(-1:1)
title('SET8, 854/28')
view(-53,48)

5. [854/32]

The points at which a helix intersects a sphere are determined.
syms t x y z phi theta
helix = [sin(t), cos(t), t]
helix = 
sphere = x^2 + y^2 + z^2 == 5
sphere = 
eq = simplify(subs(sphere, [x y z], helix)) % Substitution and solution in one line!
eq = 
P1 = subs(helix, t, -2)
P1 = 
p1 = double(P1)
p1 = 1×3
-0.9093 -0.4161 -2.0000
P2 = subs(helix, t, 2)
P2 = 
p2 = double(P2)
p2 = 1×3
0.9093 -0.4161 2.0000
% Illustration
figure
fplot3(sin(t), cos(t), t, [-4 4], 'b', 'LineWidth', 3); hold on
s = sqrt(5)
s = 2.2361
fsurf(s*sin(phi)*cos(theta), s*sin(phi)*sin(theta), s*cos(phi), [0, pi, 0, 2*pi], ...
'g', 'MeshDensity', 16)
plot3(p1(1), p1(2), p1(3), 'ro', 'MarkerFaceColor', 'r', 'MarkerSize', 12)
plot3(p2(1), p2(2), p2(3), 'ro', 'MarkerFaceColor', 'r', 'MarkerSize', 12)
alpha 0.2
axis equal; xlabel('x'); ylabel('y'); zlabel('z')
title('SET8, 854/32')

6. [854/36]

The curves lies on the unit sphere at the origin (since ).
syms t phi theta
x = cos(8*cos(t))*sin(t), y = sin(8*cos(t))*sin(t), z = cos(t)
x = 
y = 
z = 
figure
fsurf(sin(phi)*cos(theta), sin(phi)*sin(theta), cos(phi), [0, pi, 0, 2*pi], ...
'y', 'MeshDensity', 8, 'EdgeColor', 'none'); hold on
alpha 0.2
fplot3(x,y,z, [0 2*pi], 'm', 'LineWidth', 3)
axis equal; axis([-1 1 -1 1 -1 1])
xticks(-1:1); yticks(-1:1); zticks(-1:1)
xlabel('x'); ylabel('y'); zlabel('z')
title('SET8, 854/36')

7. [854/40]

The curves also lies on the unit sphere at the origin (since ).
syms t phi theta
x = sqrt(1 - cos(10*t)^2 / 4) * cos(t)
x = 
y = sqrt(1 - cos(10*t)^2 / 4) * sin(t)
y = 
z = cos(10*t) / 2
z = 
figure
fsurf(sin(phi)*cos(theta), sin(phi)*sin(theta), cos(phi), [0, pi, 0, 2*pi], ...
'r', 'MeshDensity', 8, 'EdgeColor', 'none'); hold on
alpha 0.2
fplot3(x,y,z, [0 2*pi], 'b', 'LineWidth', 3)
axis equal; axis([-1 1 -1 1 -1 1])
xticks(-1:1); yticks(-1:1); zticks(-1:1)
xlabel('x'); ylabel('y'); zlabel('z')
title('SET8, 854/40')

8. [855/46]

The circular cylinder and the semiellipsolid intersect in two semi-elliptical arcs.
syms r t x y z phi theta
surf1 = x^2 + z^2 == 1
surf1 = 
surf2 = x^2 + y^2 + 4*z^2 == 4
surf2 = 
c = solve([surf1, surf2], [x y z], 'ReturnConditions', true)
c = struct with fields:
x: [4×1 sym] y: [4×1 sym] z: [4×1 sym] parameters: [1×1 sym] conditions: [4×1 sym]
xt = simplify(subs(c.x, t)), yt = simplify(subs(c.y, t)), zt = subs(c.z, t)
xt = 
yt = 
zt = 
% We have the curve of intersection parametrically specified in 2 of 4 pieces.
% We can now illustrate by plotting the surfaces and the curve.
figure
fsurf(cos(theta), y, sin(theta), [0 2*pi -3 3], 'r', 'EdgeColor', 'k', ...
'MeshDensity', 20); hold on
fsurf(2*sin(phi)*cos(theta), 2*sin(phi)*sin(theta), cos(phi), [0 pi 0 pi], 'g', ...
'EdgeColor', 'k', 'MeshDensity', 20); alpha 0.2
fplot3(xt(1), yt(1), zt(1), [-1 1], 'm', 'LineWidth', 4)
% fplot3(xt(2), yt(2), zt(2), [-1 1], 'm', 'LineWidth', 4)
fplot3(xt(3), yt(3), zt(3), [-1 1], 'm', 'LineWidth', 4)
% fplot3(xt(4), yt(4), zt(4), [-1 1], 'm', 'LineWidth', 4)
axis equal
xlabel('x'); ylabel('y'); zlabel('z')
title('SET8, 855/46')
view(146,14)

9. [855/48]

The parabolic cylinder and the semiellipsolid intersect in two quarter-elliptical arcs.
syms r t x y z phi theta
surf1 = x^2 + 4*y^2 + 4*z^2 == 16
surf1 = 
surf2 = y == x^2
surf2 = 
c = solve([surf1, surf2], [z y x], 'ReturnConditions', true)
c = struct with fields:
z: [2×1 sym] y: [2×1 sym] x: [2×1 sym] parameters: [1×1 sym] conditions: [2×1 sym]
xt = simplify(subs(c.x, t)), yt = simplify(subs(c.y, t)), zt = subs(c.z, t)
xt = 
yt = 
zt = 
% We have the curve of intersection parametrically specified in two pieces.
% We can now illustrate by plotting the surfaces and the curve.
figure
fsurf(4*sin(phi)*cos(theta), 2*sin(phi)*sin(theta), 2*cos(phi), [0 pi/2 0 2*pi], 'c', ...
'EdgeColor', 'k', 'MeshDensity', 20); hold on
fsurf(x, x^2, z, [-2 2 -3 3], 'y', 'EdgeColor', 'k', ...
'MeshDensity', 20); alpha 0.2
% Ascertain t-range.
t_range_appx = double(solve(sqrt(16 - t^2 - 4*t^4), t)) % Throw out complex values.
t_range_appx = 4×1 complex
0.0000 + 1.4591i 1.3707 + 0.0000i 0.0000 - 1.4591i -1.3707 + 0.0000i
fplot3(xt(2), yt(2), zt(2), [-1.3707 1.3707], 'm', 'LineWidth', 4)
axis equal
xlabel('x'); ylabel('y'); zlabel('z')
title('SET8, 855/48')
view(-193,32)

10. [855/50]

While the particles do not collide (same place at same time), their paths do cross.
syms s t u
r1(s) = [s s^2 s^3], r2(t) = [1+2*t 1+6*t 1+14*t]
r1(s) = 
r2(t) = 
eqs1 = r1(u) == r2(u)
eqs1 = 
any_sols = solve(eqs1, u) % No solutions: the particles don't collide (same place, same time).
any_sols = Empty sym: 0-by-1
eqs2 = r1(s) == r2(t)
eqs2 = 
p = solve(eqs2, [s t]) % Solutions: paths of particles cross at different times.
p = struct with fields:
s: [2×1 sym] t: [2×1 sym]
ps = p.s, pt = p.t
ps = 
pt = 
A = r1(1), B = r1(2)
A = 
B = 
A = r2(0), B = r2(1/2)
A = 
B = 
% Illustration
figure
fplot3(s, s^2, s^3, [0.5 2.5], 'b', 'LineWidth', 3); hold on
fplot3(1+2*t, 1+6*t, 1+14*t, [-0.5 1], 'r', 'LineWidth', 3)
plot3(1,1,1, 'go', 'MarkerFaceColor', 'g', 'MarkerSize', 12)
plot3(2,4,8, 'go', 'MarkerFaceColor', 'g', 'MarkerSize', 12)
axis([0 3 -2 6 -5 15])
xticks(0:3); yticks(-2:2:6); zticks(-5:5:15)
xlabel('x'); ylabel('y'); zlabel('z')
title('SET8, 855/50')
view(-22,26)