12.1: Spatial Vectors

The word vector means different things to different people. In MATLAB, a vector is a matrix that has either one row or one column. So far, we’ve used MATLAB vectors to represent the following:

Sequences

A mathematical sequence, like the Fibonacci numbers, is a set of values identified by integer indices; in Chapter 4.5, we used a MATLAB vector to store the elements of a sequence.

State vectors

A state vector is a set of values that describes the state of a physical system. When you callode45, you give it initial conditions in a state vector. Then, when ode45 calls your rate function, it gives you a state vector.

Time series

One of the results from ode45 is a vector that represents a sequence of time values.

In this chapter, we’ll see another use of MATLAB vectors: representing spatial vectors. A spatial vector represents a multidimensional physical quantity like position, velocity, acceleration, or force.

For example, to represent a position in two-dimensional space, we can use a vector with two elements:

>> P = [3 4]

To interpret this vector, we have to know the coordinate system it is defined in. Most commonly, we use a Cartesian system where the x-axis points east and the y-axis points north. In that case P represents a point 3 units east and 4 units north of the origin.

When a spatial vector is represented in this way, we can use it to compute the magnitude and direction of a physical quantity. For example, the magnitude of P is the distance from the origin to P, which is the hypotenuse of the triangle with sides P(1) and P(2). We can compute it using the Pythagorean theorem:

>> sqrt(P(1)^2 + P(2)^2)
ans = 5

Or we can do the same thing using the function norm, which computes the Euclidean norm of a vector, which is its magnitude:

>> norm(P)
ans = 5

There are two ways to get the direction of a vector. One convention is to compute the angle between the vector and the x-axis:

>> atan2(P(2), P(1))
ans = 0.9273

In this example, the angle is about 0.9. But for computational purposes, we often represent direction with a unit vector, which is a vector with length 1. To get a unit vector we can divide a vector by its length:

function res = hat(V)
    res = V / norm(V)
end

This function takes a vector, V, and returns a unit vector with the same direction as V. It’s called hat because in mathematical notation, unit vectors are written with a “hat” symbol. For example, the unit vector with the same direction as \(\textbf{P}\) would be written \(\hat{P}\).