Vector Calculus for Engineers
CME100, Fall 2004
Handout #1
Vectors in 2 and 3D
1. An airplane flying due east at 500 mph in still air
encounters a 70 mph tailwind acting in the direction 60o
north of east. Determine the airplane’s ground speed
and direction.
% Define vectors
% u – velocity of plane with respect
%
to wind
% v – velocity of wind with respect to
%
ground
u=[500,0]
u =
500
0
v=[70*cos(60*pi/180), 70*sin(60*pi/180)]
v =
35.0000
60.6218
% Add u and v
w=u+v
w =
535.0000
60.6218
% Find magnitude of w
speed=norm(w)
speed =
538.4236
% Determine direction
theta=atan(w(2)/w(1))*180/pi
theta =
6.4647

w


u

v
Dot and Cross Products
2. A force of 10 N is exerted by a thruster on a satellite at
45o with respect to the horizontal. Exhaust particles hit
the solar array surface producing a “drag” force of 0.5
N pointing in the direction normal to the surface of the
array.
a) Determine the loss in thrust due to drag
b) Determine the component of drag in the direction
perpendicular to the direction of thrust
y
% Define vectors
T=[10*cos(45*pi/180), 10*sin(45*pi/180)]
T =
7.0711
7.0711
f=[0,-0.5]
f =
0
-0.5000
% Find projection of f in the direction of T
proj_T_f=dot(f,T)/norm(T)*T/norm(T)
proj_T_f =
-0.2500
-0.2500
% Find magnitude of projection
norm(proj_T_f)
ans =
0.3536
% Find effective thrust level
Thrust=10-norm(proj_T_f)
Thrust =
9.6464
% Find the component of f perpendicular to T
f_perp_T=f-proj_T_f
f_perp_T =
0.2500
-0.2500
T

x
f
3. Find the area of a triangle whose vertices are A(1,0,0)
B(0,3,0) and C(0,0,2).
AB=[-1 3 0]
AB =
-1
3
0
0
2
AC=[-1 0 2]
AC =
-1
X=cross(AB,AC)
X =
6
2
3
Area = norm(X)/2
Area =
3.5000
4. An airplane is in a steady flight as shown in the
diagram. Determine the lift force produced by the tail
such that there is no net torque on the aircraft. Assume
Fw  500000 N .
Fw
Ft
2m
2m
10 m
Fg
Equations of Lines and Planes
5. Find an equation of the plane through A(0,0,1) B(2,0,0)
and C(0,3,0)
6. Find an equation of the line of intersection of the two
planes: 3 x  6 y  2 z  15 and 2 x  y  2 z  5
7. Find the shortest distance from point Q(0,1,1) to the
plane 4 y  3 z  12
Vector-Valued Functions, Projectile Motion
8. A projectile is launched from the origin at time t  0
with an initial velocity vo at an angle  with respect to
the horizontal.
a) determine the maximum height that the projectile
reaches
b) determine the flight time and range
9. For the vector-valued function:

r  a cos t i  a sin t j  bt k determine the following:
a) velocity and acceleration vectors
b) unit tangent vector
c) unit normal vector
d) curvature
e) arclength from t  0 to 2