Dot Product of Vectors Calculator
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/6/2026
The dot product of two vectors (x₁,y₁) and (x₂,y₂) is calculated with u·v = x₁×x₂ + y₁×y₂. For u=(3,4) and v=(4,−3), the dot product is exactly zero, meaning the two vectors are perpendicular (a 90° angle).
Explanation
The dot product (also called the scalar or inner product) combines two vectors into a single number, unlike other vector operations that return a new vector. Its sign and value directly reveal the relative orientation of the two vectors: a dot product of zero means the two vectors are perpendicular (orthogonal) to each other, a positive dot product indicates an acute angle between them (less than 90°), and a negative dot product an obtuse angle (greater than 90°). The exact angle between the two vectors is found by dividing the dot product by the product of their norms (their lengths), then applying the arccosine function to that result — this division normalizes the dot product so it depends only on the vectors' relative orientation, not their individual length. The dot product has many practical applications: in physics, it's used to calculate the work done by a force (the dot product of force and displacement), and in geometry, to quickly test whether two directions are perpendicular without measuring an angle directly — useful, for instance, for checking coordinates the same way our distance between two points calculator works with 2D coordinates, or for verifying that a triangle is a right triangle before applying our hypotenuse (Pythagorean theorem) calculator.
Example: u=(3,4) and v=(4,−3)
Inputs
Vector u: (3, 4). Vector v: (4, −3).
Calculation
u·v = 3×4 + 4×(−3) = 12 − 12 = 0. A dot product of zero means the two vectors are perpendicular, confirmed by the calculated angle: exactly 90°.
Result
These two vectors are perpendicular: their dot product is zero and the angle between them is exactly 90°.
Frequently asked questions
Why does a dot product of zero mean the vectors are perpendicular?
Because the dot product is proportional to the cosine of the angle between the two vectors (u·v = ‖u‖×‖v‖×cos(θ)), and the cosine of a 90° angle is exactly 0. As long as neither vector is the zero vector, the only way to make this product zero is for the angle between them to be precisely 90°.
Does this calculator also work for three-dimensional vectors?
No, this calculator only covers two-dimensional vectors (in a plane). The principle generalizes directly to three dimensions by adding a third component z to the dot product (u·v = x₁x₂ + y₁y₂ + z₁z₂) and to the norm, but this calculator remains deliberately limited to the 2D case, the most common in everyday and educational use.
What happens if one of the two vectors entered is the zero vector (0, 0)?
The angle between a zero vector and any other vector isn't mathematically defined, since a zero vector has no direction of its own. In that case, calculating the angle involves dividing by a zero norm, which the site's engine treats as an undefined value (shown as not available) rather than a blocking error or a made-up number.