LifeTally
2D vector dot product and determinant
What are your two vector products, and are their directions parallel or perpendicular?
Enter four plane-vector components to find the dot product, signed determinant and relation between nonzero directions.
Your two vectors
Use the same perpendicular Cartesian axes, equal axis scale and a consistent component unit basis. Enter decimal fractions, not a/b. All four components are required.
How the estimate works
The dot product is ux·vx + uy·vy; the signed determinant is D = ux·vy − uy·vx. For two nonzero vectors, D = 0 means parallel directions, and dot product = 0 means perpendicular directions. Otherwise neither relation holds. If either vector is zero, both products are zero, but direction comparison is unavailable. Classification uses exact products of the accepted numerical values, without a near-zero tolerance.
u = (ux, uy); v = (vx, vy)
u · v = ux·vx + uy·vy
D = ux·vy − uy·vxOnly two declared real vectors in a plane on perpendicular axes with equal scale. This does not calculate lengths, angles, projections, a 3D cross-product vector, graphs, geographic directions or unit conversions. It does not validate measurements, estimate uncertainty or guarantee a physical or safety result. Exact relations of accepted values do not prove relations between measured objects.
OpenStax: dot product and orthogonality ↗OpenStax: two-by-two determinants ↗
OpenStax: plane vectors and the zero vector ↗
Check the filled example
Our chosen example: u = (2, −1), v = (3, 4). Dot product = 2×3 + (−1)×4 = 2; D = 2×4 − (−1)×3 = 11. The directions are neither parallel nor perpendicular. These are mathematical example values, not standards or a forecast.
Accepted input values
- Vector u: x component
- 2
- Vector u: y component
- -1
- Vector v: x component
- 3
- Vector v: y component
- 4
Plan the next step
Use the standard calculator for separate arithmetic. Check that your components use the intended axes and units.
Open the standard calculator →