Dot Product Calculator - Vector Dot Product and Angle
Calculate the dot product of two vectors with each product shown, plus vector magnitudes and the angle between them - updating live as you type.
Dimension
Vector a
Vector b
Your input
Dot with .
Solution
Magnitudes and angle
Answer
a · b =
Runs entirely on your device - nothing is ever uploaded.
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How to use the dot product calculator
- Pick the dimension - 2D or 3D, or type any size up to 8.
- Type a number (decimals and fractions like 1/5 both work) into each component of both vectors.
- The solution writes itself: every product aibi appears in the sum, then the magnitudes and the angle between the vectors.
- Read the sign of the result: positive means an acute angle, zero means perpendicular, negative means obtuse.
No calculate button - solutions are always on, updating the moment you type.
Dot product and the angle
The dot product is the bridge between algebra and geometry: divide it by both magnitudes and you get cos θ, the cosine of the angle between the vectors. A result of zero flags perpendicular vectors, which is how graphics engines test surface facing and how search engines test orthogonality. With unit vectors the math is trivial - their dot product is the cosine directly, no division needed.