Enter the components of two vectors to compute their dot product (scalar product). Input Vector U and Vector V as comma-separated values (e.g. 3, -1, 2) — supporting 2D, 3D, or higher dimensions — and get the scalar result, along with each component-wise multiplication step, magnitudes, and the angle between the vectors.
Dot Product (U · V)
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Magnitude of U (|U|)
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Magnitude of V (|V|)
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Angle Between Vectors (θ)
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Orthogonal (Perpendicular)?
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The dot product of two vectors U = ⟨u₁, u₂, ..., uₙ⟩ and V = ⟨v₁, v₂, ..., vₙ⟩ is calculated as U · V = u₁v₁ + u₂v₂ + ... + uₙvₙ. You multiply each pair of corresponding components and then sum all those products together. The result is always a scalar (a single number), not a vector.
The dot product (scalar product) multiplies corresponding components of two vectors and sums them, producing a single scalar number. The cross product (vector product) produces a new vector perpendicular to both input vectors. The dot product works in any number of dimensions, while the cross product is defined specifically for 3D vectors.
The angle θ between two vectors can be found using the formula: cos(θ) = (U · V) / (|U| × |V|), where |U| and |V| are the magnitudes (lengths) of the vectors. Take the arccos of the result to get θ in degrees or radians. This calculator computes the angle automatically once you enter your vectors.
A dot product of zero means the two vectors are orthogonal, meaning they are perpendicular to each other (the angle between them is 90°). This is a fundamental concept in linear algebra and physics — for example, a force acting perpendicular to a displacement does no work, reflected by a zero dot product.
Yes. The dot product is negative when the angle between the two vectors is greater than 90° (obtuse). A negative result indicates the vectors point in generally opposite directions. A positive result means the angle is less than 90° (acute), and zero means they are perpendicular.
Absolutely. The dot product formula extends to any number of dimensions — you simply multiply each pair of corresponding components and sum the results. This calculator supports 2D, 3D, and higher-dimensional vectors as long as both vectors have the same number of components.
The dot product has wide applications in physics (calculating work done by a force), computer graphics (lighting and shading calculations), machine learning (similarity measures between feature vectors), and signal processing. It is also used to determine whether vectors are parallel or perpendicular, and to project one vector onto another.
The scalar projection of vector U onto vector V is given by (U · V) / |V|. It tells you how much of U lies in the direction of V. When this value equals the magnitude of U, the vectors are parallel. This concept is used heavily in physics to decompose forces and velocities into directional components.