Beyond basic array arithmetic, MATLAB has several functions built specifically for arrays and vectors, such as averaging a set of sensor readings, finding the magnitude of a force vector, or computing the work done by a force over a displacement.

Functions on a Single Array

The following functions operate on a single array:

Function Description MATLAB Help Center
flip Reverse the elements in an array flip
length Length of largest array dimension length
max Largest value in array max
mean Mean or average value in array mean
median Median value of an array median
min Minimum value of an array min
norm Vector norm (magnitude) of array norm
prod Product of all values in array prod
sort Sort array in ascending order sort
std Standard deviation of array std
sum Total of all values in array sum

For a vector, norm computes the magnitude, $\sqrt{x^2 + y^2 + z^2 + …}$, which you need any time you care about the size of a force, velocity, or displacement rather than its individual components.

forces = [30, 45, -10, 22]; % N, readings from a load cell

disp('Mean force (N):')
disp(mean(forces))

resultant = [30, 40, 0]; % N
disp('Magnitude of the resultant force (N):')
disp(norm(resultant))


Mean force (N):
   21.7500

Magnitude of the resultant force (N):
    50

Functions on Multiple Arrays

These functions operate on two arrays at once:

Function Description MATLAB Help Center
dot(a,b) Dot product of two vectors dot
cross(a,b) Cross product of two vectors cross

The dot product of two vectors of the same length returns a single number, and is used throughout engineering to compute things like the work done by a force. The cross product is only defined for vectors of length 3; it returns a new vector perpendicular to both inputs, and is used to compute things like torque.

Example: Torque and Work

Question

A technician tightens a bolt using a wrench, then pushes a cart down a hallway.

For the wrench, the lever arm from the bolt to the technician’s hand is r = [0.15, 0, 0] m, and the applied force is F = [0, 40, 0] N. Find the resulting torque vector.

For the cart, the applied force is force = [120, -40, 0] N, and the cart moves along displacement = [5, 2, 0] m. Find the work done on the cart.

Solution

Torque is the cross product of the lever arm and the applied force, while work is the dot product of the force and the displacement.

% Torque from a wrench
r = [0.15, 0, 0]; % m, lever arm from bolt to hand
F = [0, 40, 0];   % N, applied force

torque = cross(r, F);
disp('Torque vector (N*m):')
disp(torque)

% Work done pushing a cart
force = [120, -40, 0]; % N
displacement = [5, 2, 0]; % m

work = dot(force, displacement);
disp('Work done (J):')
disp(work)


Torque vector (N*m):
     0     0     6

Work done (J):
   520

Reading Questions

  1. What does norm compute for a vector, and what physical quantities is that useful for?
  2. What is the difference between what dot returns and what cross returns?
  3. If you take the dot product of a force vector and a displacement vector, what physical quantity do you get?
  4. If you take the cross product of a lever arm vector and a force vector, what physical quantity do you get?
  5. Does dot work on vectors of any length, or only vectors of length 3? What about cross?

Practice Problem: Rotor Blade Load Survey

A helicopter’s main rotor blades don’t all carry exactly the same load. Small differences in manufacturing, wear, and adjustment mean each blade’s lift force is a little different, and technicians track that spread as part of routine maintenance.

In this practice problem, you’ll write a MATLAB script that summarizes a set of blade load readings, then uses the dot product to find the power delivered by the rotor during a climb.

Your Task

Write a script named rotor_loads.m that starts from the given blade load readings:

blade_loads = [4820, 5100, 4950, 5030]; % lbf, one reading per blade

Compute:

  1. mean_load - the mean of blade_loads
  2. load_spread - the difference between the highest and lowest blade loads
  3. load_std - the standard deviation of blade_loads

Next, given the helicopter’s climb velocity and the rotor’s thrust vector:

v = [0, 0, 8]; % m/s, climb velocity
F = [150, -80, 45000]; % N, rotor thrust vector
  1. climb_power - the power, in watts, delivered by the thrust vector along the climb direction (the dot product of F and v)

Your variable names for the four answers above must match exactly (mean_load, load_spread, load_std, climb_power) so that the checker below can find them.

Checking Your Work

Download check_rotor_loads.m and save it in the same folder as your rotor_loads.m script. Make sure that folder is your Current Folder in MATLAB, then run:

>> check_rotor_loads

The checker runs your script and reports whether each of the four values is correct. This is practice, not a graded assignment. If something doesn’t pass, use the feedback to find and fix the issue, then run the checker again.