MATLAB - Vectors

MATLAB - Vectors

In MATLAB, a vector is a one-dimensional array of numbers. As MATLAB stands for Matrix Laboratory, vectors are the fundamental building block for nearly all data manipulation. Understanding vectors is the key to writing efficient, readable, and idiomatic MATLAB code through a powerful concept called vectorization.

MATLAB supports two types of vectors:

• Row Vectors: A 1xN array.
• Column Vectors: An Nx1 array.

Creating Vectors

Manual Creation

Row vectors are created by enclosing elements in square brackets [], separated by spaces or commas.

% A row vector
row_vec = [7 8 9 10 11]
% or alternatively
% row_vec = [7, 8, 9, 10, 11]

Column vectors are created similarly, but elements are separated by semicolons ;.

% A column vector
col_vec = [7; 8; 9; 10; 11]

Programmatic Creation

It’s more common to create vectors using functions:

% Colon operator for uniformly spaced elements (step of 1)
v1 = 1:5         % Result: [1 2 3 4 5]

% Colon operator with a specified step
v2 = 0:0.5:2     % Result: [0 0.5 1.0 1.5 2.0]

% `linspace` for a specific number of points
v3 = linspace(0, 10, 5) % 5 points from 0 to 10. Result: [0 2.5 5 7.5 10]

% Pre-allocating a vector with zeros (good for performance)
v4 = zeros(1, 4) % Result: [0 0 0 0]

Accessing Vector Elements (Indexing)

You can access and modify vector elements using parentheses (). MATLAB indexing starts at 1.

v = [10 20 30 40 50];

% Access the third element
elem3 = v(3)      % Result: 30

% Access the last element using the 'end' keyword
last_elem = v(end) % Result: 50

% Access a range of elements (a 'slice')
sub_vec = v(2:4)  % Result: [20 30 40]

% Access non-contiguous elements
non_contig = v([1 4]) % Result: [10 40]

% Modify an element
v(1) = 99;        % v is now [99 20 30 40 50]

Logical Indexing

A powerful feature is to access elements based on a logical condition. This is a cornerstone of vectorization.

v = [10 20 30 40 50];

% Find all elements greater than 25
large_vals = v(v > 25) % Result: [30 40 50]

Essential Vector Operations

Operations on vectors are typically performed element-by-element.

a = [1 2 3];
b = [4 5 6];

% Addition and Subtraction (vectors must be same size)
sum_vec = a + b     % Result: [5 7 9]
diff_vec = b - a    % Result: [3 3 3]

% Scalar multiplication
scaled_vec = a * 3  % Result: [3 6 9]

% Element-wise multiplication (NOTE the dot: .*)
% This is NOT a dot product.
elem_prod = a .* b  % Result: [1*4, 2*5, 3*6] -> [4 10 18]

% Transpose (converts a row to a column and vice-versa)
col_a = a'          % Result: a 3x1 column vector

% Appending vectors (concatenation)
horz_cat = [a, b]   % Result: [1 2 3 4 5 6]
vert_cat = [a'; b'] % Result: a 6x1 column vector

% Vector magnitude (Euclidean norm)
mag = norm(a)       % Result: sqrt(1^2 + 2^2 + 3^2) -> sqrt(14)

% Dot product
dp = dot(a, b)      % Result: 1*4 + 2*5 + 3*6 -> 32

Common Mistake: * vs. .*

The most common error for beginners is confusing matrix multiplication (*) with element-wise multiplication (.*).

• a .* b (Element-wise): Multiplies corresponding elements. a and b must have the same size. The result is a vector of the same size.
• a * b (Matrix Multiplication): Performs linear algebraic multiplication. For vectors, this is only defined if the inner dimensions agree (e.g., row_vec * col_vec is a scalar dot product, col_vec * row_vec is a matrix outer product). Using this by mistake often leads to a ‘dimension mismatch’ error.

Practical Application: 2D Kinematics

Vectors are perfect for representing physical quantities like position, velocity, and acceleration.

% An object's initial position and velocity vectors
pos0 = [10, 20]; % (x, y) position in meters
vel = [5, -2];   % (vx, vy) velocity in m/s

% Time elapsed
dt = 2; % seconds

% Calculate the new position using vector addition
pos1 = pos0 + vel * dt;

fprintf('New position is (%.1f, %.1f) meters.\n', pos1(1), pos1(2));

% Calculate the speed (magnitude of the velocity vector)
speed = norm(vel);
fprintf('Speed is %.2f m/s.\n', speed);