12.1: Spatial Vectors
- Page ID
- 84547
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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 = 5Or 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 = 5There 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 = 0.9273In 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)
endThis 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 \(\hat{P}\).