Last updated: 2026-07-01

The angle between two vectors

Two vectors that share a starting point open up an angle between them. That angle measures how aligned the vectors are: a small angle means they point nearly the same way, a right angle means they are perpendicular, and a straight angle means they point in opposite directions. The dot product turns this geometric idea into a formula that works straight from the components.

The formula

The dot product of two vectors relates to the angle between them through

Solving for the angle gives the working formula:

The fraction inside is the cosine of the angle. Because dividing by the magnitudes removes the lengths, only the directions matter — scaling either vector leaves the angle unchanged.

The pieces

• Dot product: $\mathbf{a} \cdot \mathbf{b} = a_x b_x + a_y b_y + a_z b_z$.
• Magnitude: $|\mathbf{a}| = \sqrt{a_x^2 + a_y^2 + a_z^2}$, and likewise for $\mathbf{b}$.
• Cosine: divide the dot product by the product of the magnitudes. The result always lands between −1 and 1.
• Angle: take the inverse cosine. The answer falls between 0° and 180°.

Worked example

Let a = (1, 0, 0) and b = (1, 1, 0).

The dot product is

The magnitudes are

So the cosine of the angle is

and the angle is

Orthogonality check

A right angle is the easiest case to spot. When the dot product is exactly zero, the cosine is zero and the angle is 90°: the vectors are perpendicular, or orthogonal. For a = (1, 0) and b = (0, 1) the dot product is $(1)(0) + (0)(1) = 0$, so the angle between them is 90°. This is why testing $\mathbf{a} \cdot \mathbf{b} = 0$ is the standard way to check whether two directions are at right angles.

Same direction and opposite directions

Two other landmark cases bound the range:

• When the vectors point the same way, the cosine is 1 and the angle is 0°.
• When they point in opposite directions, the cosine is −1 and the angle is 180°.

Every other pair sits between these extremes, which is why the angle between two vectors always lies in the interval from 0° to 180°.

Degrees or radians

The same angle can be read in either unit. Degrees split a full turn into 360 parts; radians measure the angle as arc length on a unit circle, with a full turn equal to $2\pi$. The conversion is $180° = \pi\ \text{rad}$, so 45° is $\pi/4$ and 90° is $\pi/2$. Switch the unit on the result to see whichever form your problem calls for.

2D and 3D

The formula is identical in two and three dimensions — only the number of component products changes. For a flat 2D problem, leave both z-components at 0; the dot product, the magnitudes, and the angle all come out the same as if the z terms were never there.

Where it comes up

• Physics: the work done by a force depends on the angle between the force and the displacement, $W = |\mathbf{F}|\,|\mathbf{d}|\cos\theta$.
• Computer graphics: the angle between a surface normal and a light direction sets how brightly a surface is lit.
• Navigation and geometry: finding the heading difference between two directions, or the corner angle of a shape from its edge vectors.
• Machine learning: cosine similarity between feature vectors is exactly the cosine of the angle computed here.