# MATH 251: Calculus 3, SET8

## 12: Vectors and the Geometry of Space

### 12.3: The Dot Product

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

#### 1. [812/8]

The dot product of two vectors specified via components is computed.
a = sym([3 2 -1]), b = sym([4 0 5])
a = b = a_dot_b = dot(a,b)
a_dot_b = 7

#### 2. [812/10]

The dot product of two vectors specified via magnitudes and included angle is computed.
length_of_a = 80, length_of_b = 50, theta = sym(3*pi/4)
length_of_a = 80
length_of_b = 50
theta = a_dot_b = length_of_a * length_of_b * cos(theta)
a_dot_b = #### 3. [812/12]

The dot products of vectors are computed using a figure in the statement of the problem.
length_of_u = sym(1), length_of_v = sym(1/sqrt(2)), length_of_w = 1
length_of_u = 1
length_of_v = length_of_w = 1
alpha = sym(pi/4), beta = sym(pi/2)
alpha = beta = u_dot_v = length_of_u * length_of_v * cos(alpha)
u_dot_v = u_dot_w = length_of_u * length_of_w * cos(beta)
u_dot_w = 0

#### 4. [813/22]

The angles of a triangle with specified vertices are computed.
A = [1 0 -1], B = [3 -2 0], C = [1 3 3]
A = 1×3
1 0 -1
B = 1×3
3 -2 0
C = 1×3
1 3 3
alpha = acosd(dot(B-A, C-A) / (norm(B-A) * norm(C-A)))
alpha = 97.6623
beta = acosd(dot(A-B, C-B) / (norm(A-B) * norm(C-B)))
beta = 53.5008
gamma = acosd(dot(A-C, B-C) / (norm(A-C) * norm(B-C)))
gamma = 28.8370
check = alpha + beta + gamma % Sum of angles of triangle equals 180 degrees.
check = 180

#### 5. [813/24]

Pairs of vectors are determined to be orthogonal (perpendicular), parallel, or neither.
% (a)
u = [-5 4 -2], v = [3 4 -1]
u = 1×3
-5 4 -2
v = 1×3
3 4 -1
u_dot_v = dot(u,v) % Since dot product is nonzero, the vectors are not perpendicular.
u_dot_v = 3
u_by_b = u./v % Since u is not a constant multiple of V, the vectors are not parallel.
u_by_b = 1×3
-1.6667 1.0000 2.0000
% Therefore, the vectors are neither perpendicular nor parallel.
% (b)
u = [9 -6 3], v = [-6 4 -2]
u = 1×3
9 -6 3
v = 1×3
-6 4 -2
u_by_v = u./v % Since u = 1.5*v is a constant multiple of v, u and v are parallel.
u_by_v = 1×3
-1.5000 -1.5000 -1.5000
% (c)
syms c
u = [c c c], v = [c 0 -c]
u = v = u_dot_v = dot(u,v) % Since dot product is zero, the vectors are perpendicular.
u_dot_v = 0

#### 6. [813/28]

Two unit vectors are found that make an angle of with a specified vector.
v = sym([3 4]) % the specified vector
v = u = v / norm(v) % a unit vector in the direction of the specified vector
u = alpha = atan(sym(4/3))
alpha = u_polar = [1 alpha] % polar representation of u
u_polar = w = [1 alpha+pi/3] % Rotate u clockwise 60 degrees (pi/3 radians)
w = z = [1 alpha-pi/3] % Rotate u counterclockwise 60 degrees (pi/3 radians)
z = w_rect = double(1*[cos(w(2)), sin(w(2))]) % w in rectangular components (approx)
w_rect = 1×2
-0.3928 0.9196
z_rect = double(1*[cos(z(2)), sin(z(2))]) % z in rectangular components (approx)
z_rect = 1×2
0.9928 -0.1196
w_angle = atan2d(w_rect(2), w_rect(1)) % angle in deg for w w.r.t. positive x-axis (approx)
w_angle = 113.1301
z_angle = atan2d(z_rect(2), z_rect(1)) % angle in deg for z w.r.t. positive x-axis (approx)
z_angle = -6.8699
length_of_w = norm(w_rect) % unit vector
length_of_w = 1
length_of_z = norm(z_rect) % unit vector
length_of_z = 1
% Illustrate
figure
quiver(0,0, 3/5,4/5, 'b', 'AutoScale', 'off'); grid on; hold on
quiver(0,0, w_rect(1),w_rect(2), 'r', 'AutoScale', 'off')
quiver(0,0, z_rect(1),z_rect(2), 'm', 'AutoScale', 'off')
plot([-1 1], [0 0], 'k')
plot([0 0], [-0.5 1], 'k')
axis equal; axis([-0.5 1 -0.5 1])
xlabel('x'); ylabel('y')
title('SET8, 813/28') #### 7. [813/30]

Write the lines as and . The angles these lines make with the positve x-axis are and , respectively. So the obtuse angle between the lines is . The acute angle between the lines is .
alpha = atan2d(-1,2)
alpha = -26.5651
beta = atan2d(5,1)
beta = 78.6901
gamma = beta - alpha
gamma = 105.2551
theta = 180 - gamma
theta = 74.7449
%
syms x
figure
fplot(-1/2*x + 7/2); grid on; hold on
fplot(5*x - 2)
plot([-2 2], [0 0], 'k')
plot([0 0], [-5 5], 'k')
plot(1, 3, 'mo', 'MarkerFaceColor', 'm')
axis equal; axis([-2 2 -2 5])
xlabel('x'); ylabel('y')
title('SET8, 813/30') #### 8. [813/32]

The acute angle between two curves at their point of intersection is computed.
syms x y
f(x) = sin(x), g(x) = cos(x)
f(x) = g(x) = Df(x) = diff(f(x), x), Dg(x) = diff(g(x), x)
Df(x) = Dg(x) = xi = solve(f(x) == g(x), x)
xi = mf = Df(xi), mg = Dg(xi)
mf = mg = theta = double(rad2deg(atan(mf) - atan(mg))) % angle betwee curves in degrees (approx)
theta = 70.5288
Lf(x) = expand(f(xi) + mf*(x - xi))
Lf(x) = Lg(x) = expand(g(xi) + mg*(x - xi))
Lg(x) = P = [xi, f(xi)]
P = figure
fplot(f(x), [0 pi/2], 'LineWidth', 2); grid on; hold on
fplot(g(x), [0 pi/2], 'LineWidth', 2); grid on; hold on
fplot(Lf(x), [0 pi/2], 'b--');
fplot(Lg(x), [0 pi/2], 'r--');
plot(P(1), P(2), 'mo', 'MarkerFaceColor', 'm', 'MarkerSize', 10)
axis equal; axis([0 pi/2 0 1.5])
xlabel('x'); ylabel('y')
title('SET8, 813/32') #### 9. [813/44]

Scalar and vector projections are computed.
a = sym([1 2 3]), b = sym([5 0 -1])
a = b = comp_a_b = dot(a,b) / norm(a)
comp_a_b = comp_a_b_appx = double(comp_a_b)
comp_a_b_appx = 0.5345
proj_a_b = comp_a_b * a/norm(a)
proj_a_b = proj_a_b_appx = double(proj_a_b)
proj_a_b_appx = 1×3
0.1429 0.2857 0.4286

#### 10. [813/52]

A boat sails south with the help of a wind blowing in the direction S E with magnitude 400 lb. Find the work done by the wind as the boat moves 120 ft.
Work is the dot product of force with displacement F D .
mag_F = 400; mag_D = 120; theta = 36 % degrees
theta = 36
format bank
work = mag_F * mag_D * cosd(36) % work in ft-lb
work = 38832.82
format short