Vector Dot Product Calculator
Compute the dot product of two 3D vectors by entering their X, Y, and Z components. Commonly used in physics, computer graphics, and engineering to find the angle between vectors or project one vector onto another.
About this calculator
The dot product (also called the scalar product) takes two vectors and returns a single number. For two 3D vectors **v1** = (v1x, v1y, v1z) and **v2** = (v2x, v2y, v2z), the formula is: **v1 · v2** = (v1x × v2x) + (v1y × v2y) + (v1z × v2z). The result is a scalar, not a vector. Geometrically, the dot product equals |**v1**| × |**v2**| × cos(θ), where θ is the angle between the two vectors. This means a dot product of zero indicates the vectors are perpendicular (orthogonal). A positive dot product means the angle between them is less than 90°, while a negative result means the angle is greater than 90°. The dot product is central to projections, work calculations in physics (W = F · d), and lighting models in 3D rendering.
How to use
Let **v1** = (2, 3, 1) and **v2** = (4, −1, 5). Enter v1x = 2, v1y = 3, v1z = 1, v2x = 4, v2y = −1, v2z = 5. Apply the formula: dot = (2 × 4) + (3 × −1) + (1 × 5) = 8 + (−3) + 5 = 10. The dot product is 10. Since this is positive, the angle between the two vectors is less than 90°. To find the exact angle you would compute arccos(10 / (|v1| × |v2|)), but the dot product alone already tells you the vectors point in broadly the same direction.
Frequently asked questions
What does a negative dot product tell you about two vectors?
A negative dot product means the angle between the two vectors is greater than 90° and less than 180° — they point in broadly opposite directions. The more negative the value, the closer the vectors are to being anti-parallel (pointing exactly opposite). In physics, a negative dot product for force and displacement means the force does negative work, removing energy from the system. In graphics, a negative dot product between a surface normal and a light direction means the surface faces away from the light source.
How is the dot product different from the cross product?
The dot product produces a scalar (a number), while the cross product produces a vector perpendicular to both input vectors. The dot product measures how much two vectors align — it is maximized when they are parallel and zero when perpendicular. The cross product measures how much two vectors diverge — it is zero when parallel and maximized when perpendicular. Use the dot product for projections, angles, and work; use the cross product for normals, torque, and area of a parallelogram.
How do I use the dot product to find the angle between two vectors?
Once you have the dot product, use the geometric identity **v1 · v2** = |**v1**| × |**v2**| × cos(θ) and rearrange: θ = arccos(**v1 · v2** / (|**v1**| × |**v2**|)). First compute each vector's magnitude using the square root of the sum of squared components. Then divide the dot product by the product of the magnitudes, and take the inverse cosine. The result is in radians; multiply by 180/π to convert to degrees. This method works in any number of dimensions.