Dot product and interpretation
The dot product is a scalar. Magnitudes describe the lengths of the two vectors, while the angle lies from 0° through 180° when both have a direction. A zero vector has an undefined angle.
To solve the side lengths of a right triangle, use the Pythagorean Theorem Calculator and follow its own input conventions.
The formula
Multiply corresponding components and add their products. Divide the scalar product by the two magnitudes and apply inverse cosine to recover the angle; the displayed angle is converted to degrees.
Verified worked example
For A = (1, 0, 0) and B = (0, 1, 0), the dot product is 0. Each magnitude is 1 and the angle between the vectors is 90°.
How to use this calculator
- Enter two or three real components for Vector A.
- Enter the same number of components for Vector B in the same Cartesian basis.
- Read the scalar product and magnitudes, then check whether the angle is defined.
Vector Dot Product input conventions
The dot product multiplies corresponding components and adds the products. Its sign reflects whether the angle is acute, right, or obtuse when both vectors are nonzero. The result is a scalar, whereas a cross product would be a different operation returning a vector. Order does not affect a real Cartesian dot product.
Interpreting dot product
The angle formula assumes ordinary Euclidean coordinates with perpendicular axes and the same scale on every axis. Coordinates from latitude and longitude or a nonorthogonal basis cannot simply be substituted. A zero vector has no direction, so the page reports its angle as undefined while still returning its valid dot product. Displayed degrees are rounded; the underlying trigonometric calculation uses radians.
Assumptions & limitations
What this calculation assumes
- Both vectors use the same two- or three-dimensional orthonormal Cartesian basis.
What to keep in mind
- Two or three real Cartesian components only; no complex inner product or nonorthogonal basis.
Common questions
What does a zero dot product mean?
For two nonzero vectors, it means they are perpendicular. If either vector is zero, the product is also zero but its direction and angle are undefined.
Can I enter two-dimensional vectors?
Yes. Enter two components for each vector. Three-dimensional vectors need three entries each; mixing dimensions is rejected.
Is the dot product a cross product?
No. The dot product returns a scalar and measures alignment. A three-dimensional cross product returns a perpendicular vector and is a different operation.
Sources & further reading
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