Dot Product
Compute the dot product of two 2D or 3D vectors instantly. Free online dot product calculator shows the scalar result of v·w = v1w1 + v2w2 + v3w3 for vector algebra and physics.
About This Calculator
This dot product calculator computes the scalar (dot) product of two vectors in 2D or 3D space. For vectors v = (x1,y1,z1) and w = (x2,y2,z2), the dot product is v·w = x1x2 + y1y2 + z1z2. The result is a single scalar value that represents the magnitude of one vector projected onto the other.
The dot product can also be expressed as v·w = |v||w|cos(theta), where theta is the angle between the vectors. This relationship makes the dot product fundamental for calculating angles between vectors, determining orthogonality (perpendicular vectors have dot product zero), and computing projections of one vector onto another.
Applications: In physics, work done by a force is W = F·d (the dot product of force and displacement vectors). In computer graphics, the dot product determines how light reflects off surfaces (Lambertian shading). In machine learning, cosine similarity between document vectors uses the dot product. Enter your vectors below and click Calculate Dot Product.
Frequently Asked Questions
What is the dot product of two vectors?
The dot product (also called scalar product) of two vectors v = (v1,v2,v3) and w = (w1,w2,w3) is the scalar value v·w = v1w1 + v2w2 + v3w3. It equals |v||w|cos(theta) where theta is the angle between the vectors.
What does the dot product represent geometrically?
Geometrically, the dot product measures how much two vectors point in the same direction. A positive dot product means the vectors form an acute angle (less than 90 deg), zero means they are perpendicular (90 deg), and negative means they form an obtuse angle (greater than 90 deg).
What is the dot product of perpendicular vectors?
The dot product of perpendicular vectors is always zero. For example, (1,0,0) · (0,1,0) = 0. This is because cos(90 deg) = 0, making v·w = |v||w|cos(90 deg) = 0.
What is the difference between dot product and cross product?
The dot product produces a scalar (a single number), while the cross product produces a vector perpendicular to both inputs. Dot product measures alignment (cosine of angle), while cross product measures perpendicularity (sine of angle x area of parallelogram).
Where is the dot product used in real life?
The dot product is used extensively in physics (work = force·displacement), computer graphics (lighting calculations, determining if surfaces face the light source), machine learning (cosine similarity between vectors), game development (projection and collision detection), and engineering (stress analysis, signal processing).
Can I compute the dot product for vectors of different dimensions?
Vectors must have the same number of components to compute their dot product. This calculator handles both 2D (x,y) and 3D (x,y,z) vectors. If you leave the z component empty, it defaults to 0.