Understand the subject
What does a dot product measure?
A dot product multiplies matching components and adds them into one scalar. It records alignment, not a new vector.
Use it to test perpendicularity, measure projection, or combine compatible component data.
See the structure
What the calculation is doing
1234
row 1 · column 12012
1×2 + 2×1 = 4Worked interpretation
Read the result in context
u·v=Σᵢuᵢvᵢ=‖u‖‖v‖cos θ
Interpret with care
Important boundary
A zero dot product means perpendicular only in the Euclidean coordinate setting used here.
Use the Matrix Calculator to experiment with small exact matrix operations before applying a more specialized route.
Quick guide
How to use this calculator
- Give a and b the same dimension.
- Enter their components in matching order.
- Read the exact sum of component products.
Calculation method
Multiply matching components, then add
For a and b with n components, a · b = a₁b₁ + … + aₙbₙ. The result is a scalar.
Worked example
Dot product in three dimensions
For a = ⟨1,2,3⟩ and b = ⟨4,−1,2⟩, add the three component products.
a · b = 1×4 + 2×(−1) + 3×2 = 8
Supported inputs
Precision and limits
Vector size
General vectors may contain 2–8 components; three-dimensional products require exactly three.
Entry precision
Each signed decimal may contain up to 30 digits and 15 decimal places.
Square roots
Squared magnitudes remain exact. Irrational roots and normalized components are explicitly labelled approximations.
