Find Angle Between Vectors and Lines in MATLAB

1. Angle Between Two Vectors (2D or 3D)

Use dot(), norm(), and acosd() to get the angle in degrees directly.

% Define vectors
u = [1, 2, 3];
v = [4, -1, 2];

% Compute cosine value
cosTheta = dot(u, v) / (norm(u) * norm(v));
% Clamp value to [-1, 1] to prevent roundoff errors
cosTheta = max(min(cosTheta, 1.0), -1.0);
% Compute angle in degrees and radians
thetaDeg = acosd(cosTheta);
thetaRad = acos(cosTheta);
fprintf('Angle: %.2f degrees (%.4f radians)\n', thetaDeg, thetaRad);

2. Angle Between Two 2D Lines (From Slopes)

If you have lines defined by slopes m1 and m2, compute the angle with atan2d() or atand():

% Line 1: y = 3x + 1  (m1 = 3)
% Line 2: y = -2x + 4 (m2 = -2)
m1 = 3;
m2 = -2;

% Acute angle between the two lines
thetaDeg = atand(abs((m2 - m1) / (1 + m1 * m2)));
fprintf('Acute angle between lines: %.2f degrees\n', thetaDeg);

3. Angle Between Two Lines Given by Endpoints (3D or 2D)

Convert endpoints into direction vectors by subtracting start and end points:

% Line 1 endpoints: P1 to P2
P1 = [0, 0, 0];
P2 = [2, 3, 5];

% Line 2 endpoints: Q1 to Q2
Q1 = [1, 1, 1];
Q2 = [4, 0, 2];
% Direction vectors
d1 = P2 - P1;
d2 = Q2 - Q1;
% Acute angle between lines (uses absolute value on dot product)
cosTheta = abs(dot(d1, d2)) / (norm(d1) * norm(d2));
cosTheta = min(cosTheta, 1.0);
angleDeg = acosd(cosTheta);
fprintf('Angle between 3D lines: %.2f degrees\n', angleDeg);

4. Angle Between Batches of Vectors (Matrix Operations)

To calculate angles between multiple pairs of vectors simultaneously without writing loops:

% Matrices containing N vectors (each row is a vector)
U = [1, 0, 0; 0, 1, 0; 1, 1, 0];
V = [0, 1, 0; 0, 1, 0; 1, 0, 0];

% Row-wise dot product and row-wise norms
dotProds = sum(U .* V, 2);
normU = sqrt(sum(U.^2, 2));
normV = sqrt(sum(V.^2, 2));
cosAngles = dotProds ./ (normU .* normV);
cosAngles = max(min(cosAngles, 1.0), -1.0);
anglesDeg = acosd(cosAngles);
disp('Angles in degrees:');
disp(anglesDeg);

Summary of Built-in MATLAB Functions

       
• dot(u, v): Calculates vector dot product.

       
• norm(u): Calculates vector magnitude.

       
• acosd(x): Returns inverse cosine in degrees.

       
• acos(x): Returns inverse cosine in radians.

       
• atan2d(y, x): Returns four-quadrant inverse tangent in degrees.