Your result
Click Calculate
Enter your values, then click Calculate result.How this calculator helps
This vector calculator reports several linked but distinct operations on two three-dimensional Cartesian vectors. Dot product is a scalar, cross product is a vector, and projection is a vector along the second input. The page makes that order explicit and handles zero-vector cases without assigning a false angle. Use it for linear algebra or mechanics arithmetic with a stated coordinate frame rather than inferring physical quantities from six unlabeled numbers.
How to use it
- 1
Enter both vectors in one right-handed Cartesian frame.
- 2
Use consistent component units and zero z values for planar vectors.
- 3
Click Calculate result to obtain dot, cross, angle and projection outputs.
- 4
Check operation order and zero-vector exceptions before attaching a physical interpretation.
Formula and methodology
A·B = AxBx + AyBy + AzBz. A×B = (AyBz−AzBy, AzBx−AxBz, AxBy−AyBx). cosθ = (A·B)/(|A||B|). projB A = [(A·B)/|B|²]B.
The calculator applies the displayed arithmetic to the values entered on this device. It does not silently load a local tax rate, currency conversion or commercial assumption.
Worked calculation example
Let A = (1,0,0) and B = (0,1,0). Their dot product is zero, their cross product A×B is (0,0,1), both magnitudes are one and their angle is 90 degrees. The projection of A onto B is (0,0,0). Reversing the vector order changes the cross product to (0,0,−1) but leaves the dot product and angle unchanged. These simple coordinate-axis values make the order and direction conventions easy to verify.
How to interpret your result
Read each output according to its type. A zero dot product indicates perpendicularity only when both magnitudes are nonzero, while a zero cross product also occurs for parallel vectors or zero inputs. The projection is a directed component along B, not a distance between endpoints. These distinctions help avoid attaching an unsupported physical meaning to a correct numerical calculation.
For different inputs or formulas, use Cartesian Distance Calculator; Work Energy Calculator; Point Mass Center Of Mass Calculator.
Related questions this calculator covers
- vector dot product calculator
- cross product calculator
- vector projection calculator
Scenario comparison
| Scenario | What it shows |
|---|---|
| Coordinate axes | x and y unit vectors have zero dot and positive z cross. |
| Parallel pair | cross product vanishes and a positive alignment has zero angle. |
| Zero B | dot and cross remain calculable but angle and projection are undefined. |
Common mistakes to avoid
- Swapping the cross-product order without changing its sign.
- Assigning an angle to a zero vector.
- Using geographic coordinates as though they were Cartesian vector components.
Check the coordinate-axis example by hand. Reverse input order and confirm cross product negates while dot stays fixed. Compute the dot of the cross result with each original vector and verify zero within rounding. For nonzero B, subtract the projection from A and check its dot with B is zero. Test zero vectors and confirm undefined angle and projection cases are labeled explicitly.
Authoritative reference. Reviewed on 6 October 2026 for the specific disclosed method or measurement context. The arithmetic is independently implemented. A linked reference does not certify an individual result or extend the worksheet beyond its stated scope.What can affect the result?
Coordinates need a common frame
Enter x, y and z components for both vectors in one Cartesian frame with consistent axis orientation. For planar vectors, use zero z components. The cross-product direction assumes the usual right-handed coordinate convention. Components from differently rotated coordinate frames need transformation before comparison. Negative components are valid signed directions; they are not negative vector magnitudes. The calculator cannot identify the physical frame or convert axes from a diagram.
The dot product is a scalar
Dot product sums matching component products and is commutative: A·B equals B·A. A zero dot product between two nonzero vectors means they are perpendicular under the Euclidean model. A dot product can be negative when vectors form an obtuse angle. Its units, when physical vector units are supplied, are the product of the two input units. The worksheet keeps the numeric result generic rather than calling every dot product energy or work.
The cross product depends on order
A×B is perpendicular to both nonzero nonparallel vectors and follows the right-hand orientation. Swapping the vectors negates it. Equal or parallel vectors have a zero cross product even if they have nonzero magnitudes. The component expression is shown in the formula so signs can be checked directly. A scalar dot result and a three-coordinate cross result are different output types; one cannot be substituted for the other in a physical equation.
Magnitudes are Euclidean lengths
Each magnitude is the square root of the sum of squared components and is nonnegative. Scaling a vector by a positive factor scales its magnitude by that factor; a negative factor also reverses direction. A magnitude of zero occurs only for a zero vector. The numeric vector components are bounded and finite, and ordinary browser precision applies. Coordinate units remain those supplied by the user rather than being inferred as metres, newtons or another quantity.
Angle requires two nonzero vectors
The angle uses arccos of the dot product divided by the product of magnitudes and is reported from zero through 180 degrees. The ratio is clamped to the mathematical interval −1 through 1 to avoid tiny floating-point overshoots. If either vector is zero, its direction is undefined and the angle is explicitly marked undefined. Returning zero degrees in that case would imply an alignment that no zero-vector direction establishes.
Projection order is stated
The projection output is the vector projection of A onto B. It lies along B and equals [(A·B)/|B|²] times B. It is not the projection of B onto A and is not only the signed scalar component. If B is zero, the projection is undefined; if A is zero and B is nonzero, the projection is the zero vector. The sign of the dot product determines whether the projected vector points with or against B.
Physical interpretation needs compatible quantities
The arithmetic applies to Euclidean vectors. A force–displacement dot product can have work units, while a position–force cross product can have torque units, but those meanings require the appropriate input definitions and order. This page does not infer them. Geographic latitude and longitude are not Cartesian vector components suitable for this worksheet without a defined conversion. Keep any physical units and reference frame with the results when using them in a larger problem.
Privacy and browser processing
Values entered on this page are processed in the current browser session. SolvePilot does not require an account and does not receive the values entered into the calculator. Refreshing or closing the page clears the working values unless the browser itself restores a previous session. Avoid entering identifying or account information because the calculation needs summary values only.
Accuracy and verification
Accuracy depends first on input quality. Confirm definitions, scales, dates and source information before entering a value. Keep an independent record of any result used for planning because this page does not create an official statement or retain a calculation history.
Limits of this estimate
Swapping the cross-product order without changing its sign. Read each output according to its type. A zero dot product indicates perpendicularity only when both magnitudes are nonzero, while a zero cross product also occurs for parallel vectors or zero inputs. The projection is a directed component along B, not a distance between endpoints. These distinctions help avoid attaching an unsupported physical meaning to a correct numerical calculation. Entries remain on this device for the calculation and are not submitted to a calculation server. Browser floating-point arithmetic and displayed rounding do not establish real-world measurement certainty.
Sources and review information
Frequently asked questions
Is the dot product a vector?＋
No. It is a scalar; the cross product has three coordinates.
What happens when vector order is reversed?＋
The cross product changes sign, while dot product and angle remain the same.
Can I enter 2D vectors?＋
Yes. Set both z components to zero.
What is the angle with a zero vector?＋
It is undefined because the zero vector has no direction.
Which projection is shown?＋
The vector projection of A onto B.