Dot Product Calculator

Math Linear Algebra • Vectors and Basic Operations

Written by STEM Calculators Team Published February 21, 2026 Updated February 24, 2026

Compute the dot product \(\mathbf{A}\cdot\mathbf{B}=\sum_i a_i b_i\), the angle \(\theta=\cos^{-1}\!\left(\frac{\mathbf{A}\cdot\mathbf{B}}{\lVert\mathbf{A}\rVert\lVert\mathbf{B}\rVert}\right)\), orthogonality, and the projection of \(\mathbf{A}\) onto \(\mathbf{B}\).

Angle + projection visualization

2D: drag to pan, wheel to zoom. 3D: drag to rotate, Shift+drag to pan, wheel to zoom. “Play” rotates \(\mathbf{B}\) once (one full turn) and stops automatically.

\(\mathbf{A}\) \(\mathbf{B}\) \(\mathrm{proj}_{\mathbf{B}}(\mathbf{A})\) (dashed) angle arc

Enter vectors and click “Calculate”.

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Theory

Dot product in \(\mathbb{R}^n\)

For vectors \(\mathbf{A}=(a_1,\dots,a_n)\) and \(\mathbf{B}=(b_1,\dots,b_n)\), the dot product is \[ \mathbf{A}\cdot\mathbf{B}=\sum_{i=1}^n a_i b_i. \] It measures how much the vectors “align.”

Connection to angle

The dot product also satisfies \[ \mathbf{A}\cdot\mathbf{B}=\lVert\mathbf{A}\rVert\,\lVert\mathbf{B}\rVert\cos\theta, \] so (when both vectors are nonzero), \[ \theta=\cos^{-1}\!\left(\frac{\mathbf{A}\cdot\mathbf{B}}{\lVert\mathbf{A}\rVert\lVert\mathbf{B}\rVert}\right). \]

If \(\mathbf{A}\cdot\mathbf{B}>0\), then \(0^\circ<\theta<90^\circ\) (acute).
If \(\mathbf{A}\cdot\mathbf{B}=0\), then \(\theta=90^\circ\) (orthogonal).
If \(\mathbf{A}\cdot\mathbf{B}<0\), then \(90^\circ<\theta<180^\circ\) (obtuse).

If \(\lVert\mathbf{A}\rVert=0\) or \(\lVert\mathbf{B}\rVert=0\), the angle is undefined (zero vector has no direction).

Projection (scalar and vector)

The scalar projection (“component”) of \(\mathbf{A}\) onto \(\mathbf{B}\) is \[ \mathrm{comp}_{\mathbf{B}}(\mathbf{A})=\frac{\mathbf{A}\cdot\mathbf{B}}{\lVert\mathbf{B}\rVert}, \] and the vector projection is \[ \mathrm{proj}_{\mathbf{B}}(\mathbf{A}) =\left(\frac{\mathbf{A}\cdot\mathbf{B}}{\lVert\mathbf{B}\rVert^2}\right)\mathbf{B}. \]

Geometrically, \(\mathrm{proj}_{\mathbf{B}}(\mathbf{A})\) is the “shadow” of \(\mathbf{A}\) on the line spanned by \(\mathbf{B}\).

Physics example: work

Work done by a constant force is a dot product: \[ W=\mathbf{F}\cdot\mathbf{d}=\lVert\mathbf{F}\rVert\,\lVert\mathbf{d}\rVert\cos\theta. \] Only the component of force in the direction of displacement contributes to work.

University note: inner products

In advanced linear algebra, the dot product is generalized to an inner product (possibly for functions or complex vectors). Many ideas remain the same: length, angle, orthogonality, and projection.

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