Angle Between Two Vectors Calculator

The angle $\theta$ between two vectors comes from the dot product:

$\cos\theta = \frac{\vec{u} \cdot \vec{v}}{\|\vec{u}\|\,\|\vec{v}\|} \qquad\Longrightarrow\qquad \theta = \arccos\!\left(\frac{\vec{u} \cdot \vec{v}}{\|\vec{u}\|\,\|\vec{v}\|}\right)$

In 2D, with $\vec{u} = (u_1, u_2)$ and $\vec{v} = (v_1, v_2)$:

$\vec{u} \cdot \vec{v} = u_1v_1 + u_2v_2, \qquad \|\vec{u}\| = \sqrt{u_1^2 + u_2^2}$

In 3D the pattern just gains a term:

$\vec{u} \cdot \vec{v} = u_1v_1 + u_2v_2 + u_3v_3, \qquad \|\vec{u}\| = \sqrt{u_1^2 + u_2^2 + u_3^2}$

The formula requires both vectors to be non-zero — the zero vector has no direction, so the angle is undefined. It returns the smaller angle between the two directions, always in $[0^\circ, 180^\circ]$.

1. Dot product: multiply matching components and add.
2. Magnitudes: square each component, sum, take the square root.
3. Divide: $\cos\theta = (\vec{u}\cdot\vec{v})/(\|\vec{u}\|\|\vec{v}\|)$. This value always lies in $[-1, 1]$; anything outside means an arithmetic slip.
4. Inverse cosine: $\theta = \arccos(\cos\theta)$. Set your calculator to degrees or radians deliberately — $\pi$ radians $= 180^\circ$.

What the sign tells you immediately

Angle between two points

Points have no direction, so first convert to vectors. For the angle at vertex $B$ in triangle $ABC$, build $\vec{BA} = A - B$ and $\vec{BC} = C - B$ — both pointing away from the vertex — then apply the same formula. Subtracting in the wrong direction gives the supplement, $180^\circ - \theta$.

Direction angle of a single vector

For one 2D vector measured from the positive $x$-axis, use $\theta = \arctan(u_2/u_1)$ and adjust by $180^\circ$ when $u_1 < 0$.

• Forgetting to divide by the magnitudes. $\arccos(\vec{u}\cdot\vec{v})$ alone is meaningless and usually out of domain.
• Adding components instead of multiplying them. The dot product is $u_1v_1 + u_2v_2$, not $(u_1+v_1) + (u_2+v_2)$.
• Using the cross product in 2D. $\sin\theta = \|\vec{u}\times\vec{v}\|/(\|\vec{u}\|\|\vec{v}\|)$ works, but $\arcsin$ cannot distinguish $\theta$ from $180^\circ - \theta$. The dot product is unambiguous over the full range.
• Rounding $\cos\theta$ too early. Keep four or five decimals before applying $\arccos$; near $0^\circ$ or $180^\circ$ the inverse cosine is extremely sensitive.
• Ignoring degree/radian mode. An answer of $0.28$ is radians, $16.26$ is degrees — same angle.
• Using position vectors when you need displacement vectors. For an angle at a point, subtract that point from the other two first.