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Home › Math & Conversions › Core Math & Algebra › Dot Product Calculator Dot Product Calculator This dot product calculator works with 2D, 3D, and general n-dimensional vectors. Enter the components of two vectors and get the scalar product a·b , the length of each vector , the angle between them , and the projection of A onto B , with step-by-step expansion. It is designed for students and professionals in linear algebra, physics, engineering, and data science who need a precise and readable breakdown of vector computations. 1. Enter vectors Use comma or space separated values. The two vectors must have the same dimension. Examples: 1, 2, 3 , 0.5 -1.2 4.0 , 2 0 -3 5 . Vector A components Vector B components Decimal places Example 2D (orthogonal) Example 3D Clear inputs Calculate Clear results Supports any dimension with exact component product 2. Results Vector summary Dimension Dot product a·b |A| |B| cos(θ) Angle θ Orthogonality test Parallel
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anti-parallel Projection of A onto B Projection vector proj B (A): Length of projection |proj B (A)|: Projection is only defined when B is non-zero. If B is zero, direction is undefined. Step-by-step expansion (component form) Copy as plain text Dot product – algebraic and geometric definitions For two vectors \( \mathbf{a}, \mathbf{b} \in \mathbb{R}^n \) with components \( \mathbf{a} = (a_1, a_2, \dots, a_n) \) and \( \mathbf{b} = (b_1, b_2, \dots, b_n) \), the dot product (or scalar product ) is defined as \[ \mathbf{a} \cdot \mathbf{b} = \sum_{i=1}^{n} a_i b_i = a_1 b_1 + a_2 b_2 + \dots + a_n b_n. \] Geometrically, the dot product relates to the angle \( \theta \) between the two vectors: \[ \mathbf{a} \cdot \mathbf{b} = \|\mathbf{a}\| \, \|\mathbf{b}\| \cos\theta, \] where \( \|\mathbf{a}\| \) and \( \|\mathbf{b}\| \) denote the Euclidean norms (lengths). This gives \[ \cos\theta = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\| \, \|\mathbf{b}\|}, \qquad \theta = \arccos\!\left(\frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\| \, \|\mathbf{b}\|}\right). \] Orthogonality and angle between vectors Orthogonal vectors : \( \mathbf{a} \) and \( \mathbf{b} \) are orthogonal if and only if \( \mathbf{a} \cdot \mathbf{b} = 0 \). Geometrically, the angle between them is \( 90^\circ \). Parallel vectors : If \( \theta = 0^\circ \) or \( 180^\circ \), the vectors are parallel or anti-parallel. In practice we test whether the unit vectors have the same or opposite direction within a tolerance. Zero vector : If one of the vectors is zero, its direction is undefined, so the angle is also undefined even though the dot product is 0. Projection of one vector onto another The projection of \( \mathbf{a} \) onto a non-zero vector \( \mathbf{b} \) is the component of \( \mathbf{a} \) that lies in the direction of \( \mathbf{b} \). It is given by \[ \operatorname{proj}_{\mathbf{b}}(\mathbf{a}) = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{b}\|^2}\,\mathbf{b}. \] The scalar factor \( (\mathbf{a} \cdot \mathbf{b})

anti-parallel Projection of A onto B Projection vector proj B (A): Length of projection |proj B (A)|: Projection is only defined when B is non-zero. If B is zero, direction is undefined. Step-by-step expansion (component form) Copy as plain text Dot product – algebraic and geometric definitions For two vectors \( \mathbf{a}, \mathbf{b} \in \mathbb{R}^n \) with components \( \mathbf{a} = (a_1, a_2, \dots, a_n) \) and \( \mathbf{b} = (b_1, b_2, \dots, b_n) \), the dot product (or scalar product ) is defined as \[ \mathbf{a} \cdot \mathbf{b} = \sum_{i=1}^{n} a_i b_i = a_1 b_1 + a_2 b_2 + \dots + a_n b_n. \] Geometrically, the dot product relates to the angle \( \theta \) between the two vectors: \[ \mathbf{a} \cdot \mathbf{b} = \|\mathbf{a}\| \, \|\mathbf{b}\| \cos\theta, \] where \( \|\mathbf{a}\| \) and \( \|\mathbf{b}\| \) denote the Euclidean norms (lengths). This gives \[ \cos\theta = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\| \, \|\mathbf{b}\|}, \qquad \theta = \arccos\!\left(\frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\| \, \|\mathbf{b}\|}\right). \] Orthogonality and angle between vectors Orthogonal vectors : \( \mathbf{a} \) and \( \mathbf{b} \) are orthogonal if and only if \( \mathbf{a} \cdot \mathbf{b} = 0 \). Geometrically, the angle between them is \( 90^\circ \). Parallel vectors : If \( \theta = 0^\circ \) or \( 180^\circ \), the vectors are parallel or anti-parallel. In practice we test whether the unit vectors have the same or opposite direction within a tolerance. Zero vector : If one of the vectors is zero, its direction is undefined, so the angle is also undefined even though the dot product is 0. Projection of one vector onto another The projection of \( \mathbf{a} \) onto a non-zero vector \( \mathbf{b} \) is the component of \( \mathbf{a} \) that lies in the direction of \( \mathbf{b} \). It is given by \[ \operatorname{proj}_{\mathbf{b}}(\mathbf{a}) = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{b}\|^2}\,\mathbf{b}. \] The scalar factor \( (\mathbf{a} \cdot \mathbf{b})

Calculators in anti-parallel Projection of A onto B Projection vector proj B (A): Length of projection |proj B (A)|: Projection is only defined when B is non-zero. If B is zero, direction is undefined. Step-by-step expansion (component form) Copy as plain text Dot product – algebraic and geometric definitions For two vectors \( \mathbf{a}, \mathbf{b} \in \mathbb{R}^n \) with components \( \mathbf{a} = (a_1, a_2, \dots, a_n) \) and \( \mathbf{b} = (b_1, b_2, \dots, b_n) \), the dot product (or scalar product ) is defined as \[ \mathbf{a} \cdot \mathbf{b} = \sum_{i=1}^{n} a_i b_i = a_1 b_1 + a_2 b_2 + \dots + a_n b_n. \] Geometrically, the dot product relates to the angle \( \theta \) between the two vectors: \[ \mathbf{a} \cdot \mathbf{b} = \|\mathbf{a}\| \, \|\mathbf{b}\| \cos\theta, \] where \( \|\mathbf{a}\| \) and \( \|\mathbf{b}\| \) denote the Euclidean norms (lengths). This gives \[ \cos\theta = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\| \, \|\mathbf{b}\|}, \qquad \theta = \arccos\!\left(\frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\| \, \|\mathbf{b}\|}\right). \] Orthogonality and angle between vectors Orthogonal vectors : \( \mathbf{a} \) and \( \mathbf{b} \) are orthogonal if and only if \( \mathbf{a} \cdot \mathbf{b} = 0 \). Geometrically, the angle between them is \( 90^\circ \). Parallel vectors : If \( \theta = 0^\circ \) or \( 180^\circ \), the vectors are parallel or anti-parallel. In practice we test whether the unit vectors have the same or opposite direction within a tolerance. Zero vector : If one of the vectors is zero, its direction is undefined, so the angle is also undefined even though the dot product is 0. Projection of one vector onto another The projection of \( \mathbf{a} \) onto a non-zero vector \( \mathbf{b} \) is the component of \( \mathbf{a} \) that lies in the direction of \( \mathbf{b} \). It is given by \[ \operatorname{proj}_{\mathbf{b}}(\mathbf{a}) = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{b}\|^2}\,\mathbf{b}. \] The scalar factor \( (\mathbf{a} \cdot \mathbf{b}).

Dot Product Calculator

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