What formula is used to find the angle between two vectors?

The standard formula is cos(theta) = (A dot B) / (|A| times |B|), then theta = arccos of that value. The dot product A dot B equals the sum of the products of corresponding components. Dividing by the product of the magnitudes gives the cosine of the angle, and arccos recovers the angle itself. The result is always in the range 0 to 180 degrees (0 to pi radians).

What is the difference between the dot product and the cross product?

The dot product A dot B = |A||B|cos(theta) is a scalar — it measures how much the two vectors point in the same direction. The cross product A cross B = |A||B|sin(theta) * n-hat is a 3-D vector perpendicular to both A and B (only defined in 3-D). When the dot product is zero the vectors are perpendicular; when the cross product is zero they are parallel.

How does a negative dot product affect the angle?

A negative dot product means cos(theta) is negative, which means the angle is greater than 90 degrees (obtuse). A dot product of zero means the vectors are perpendicular (90 degrees). A positive dot product means the angle is acute (less than 90 degrees). This is a quick mental check: just compute A dot B and check its sign.

What is the signed angle in 2-D?

The arccos formula always returns an angle between 0 and 180 degrees and cannot tell you whether B is to the left or right of A. In 2-D, the signed angle uses atan2(cross, dot) to return a value in the range -180 to 180 degrees, where positive means A rotates counter-clockwise to reach B and negative means clockwise.

Can I use this for 4-D vectors?

Yes. The dot-product formula generalises to any number of dimensions: cos(theta) = (sum of ai*bi) / (|A| * |B|). The cross product is not defined in 4-D (it requires exactly 3 dimensions), so only the angle, magnitudes, dot product, and projections are shown for 4-D inputs.

What is a vector projection and how is it calculated?

The vector projection of A onto B is the component of A that lies along B. It equals (A dot B / |B|squared) times B. Geometrically it is the shadow A casts on B when light shines perpendicular to B. The scalar projection (a single number) is A dot B divided by |B|. Both are shown in the results panel.

Angle Between Vectors Calculator

The angle between two vectors is one of the most fundamental calculations in linear algebra, physics, and computer graphics. Whether you are checking whether two forces are perpendicular, computing the diffuse lighting term in a shader, analysing the angle of attack on an aerofoil, or verifying that two molecule bonds meet at the correct geometry, the formula is the same: take the dot product, divide by the product of the magnitudes, and apply arccos.

This calculator handles 2-D, 3-D, and 4-D vectors. Enter the components of each vector as comma-separated numbers and you get the angle in both degrees and radians, the cosine, the dot product, magnitudes, and a complete step-by-step derivation. For 2-D inputs the calculator also shows the signed angle (positive = counter-clockwise from A to B, negative = clockwise), which the plain arccos formula cannot distinguish. For 3-D inputs it adds the cross product A × B and its magnitude, which equals |A||B|sinθ — handy for finding the normal to a plane or the torque arm.

How it works

Given vectors A = (a₁, a₂, …) and B = (b₁, b₂, …) of the same dimension n:

cos θ = (A · B) / (|A| |B|)

where the dot product A · B = ∑ aᵢ bᵢ and the magnitudes |A| = √(∑ aᵢ²), |B| = √(∑ bᵢ²). The angle θ is then:

θ = arccos( A · B / (|A| |B|) )

The calculator clamps the cosine to the interval [-1, 1] before calling arccos so that tiny floating-point rounding errors (e.g. a value of 1.0000000002 from perfect parallel vectors) do not produce a NaN. The result is always in [0, π] (i.e. 0° to 180°).

2-D signed angle

For 2-D vectors the signed angle from A to B is computed via atan2(cross, dot) where cross = a₁b₂ - a₂b₁ (the z-component of the 3-D cross product). This returns a value in (-180°, 180°]: positive if B is counter-clockwise from A, negative if clockwise.

3-D cross product

For 3-D vectors the cross product A × B is:

(a₂b₃ - a₃b₂,  a₃b₁ - a₁b₃,  a₁b₂ - a₂b₁)

Its magnitude |A × B| = |A| |B| sin θ provides an independent check on the angle. When the cross product is the zero vector the two vectors are parallel (or anti-parallel).

Vector projections

The vector projection of A onto B is proj_B(A) = (A · B / |B|²) B. It is the portion of A that points in the direction of B. The scalar projection is A · B / |B|.

Worked example

Find the angle between A = (1, 2, 2) and B = (2, 1, -2).

|A| = √(1 + 4 + 4) = √9 = 3
|B| = √(4 + 1 + 4) = √9 = 3
A · B = (1)(2) + (2)(1) + (2)(-2) = 2 + 2 - 4 = 0
cos θ = 0 / (3 × 3) = 0
θ = arccos(0) = 90° (the vectors are perpendicular)

Cross product: A × B = (2·(-2) - 2·1, 2·2 - 1·(-2), 1·1 - 2·2) = (-6, 6, -3). |A × B| = √(36+36+9) = √81 = 9, and |A||B|sin(90°) = 3·3·1 = 9. The two routes agree.

A	B	Angle
(1, 0)	(0, 1)	90°
(1, 1)	(1, 0)	45°
(1, 0, 0)	(0, 1, 0)	90°
(1, 2, 2)	(2, 1, -2)	90°
(1, 0, 0)	(-1, 0, 0)	180°

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