Enter two vectors in 2D or 3D to get the dot product, cross product, magnitudes, the angle between them, projections and more, each with the formula and the numbers plugged in.
| Operation | Formula | Result |
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Every common vector operation from one pair of inputs.
The formula and the substituted numbers for each result.
Switch dimensions and see the vectors drawn.
The dot product A·B multiplies matching components and adds them. It is a single number equal to |A||B|cos θ, so it is zero for perpendicular vectors and is used to find angles and projections.
The cross product A×B exists in 3D and gives a vector perpendicular to both A and B, with length |A||B|sin θ, which is the area of the parallelogram they span. In 2D the calculator gives the z-component a₁b₂ − a₂b₁, whose sign tells you whether B turns counter-clockwise or clockwise from A.
The length of a vector: the square root of the sum of squared components.
Same direction, length 1, found by dividing by the magnitude.
How much of A points along B.
Their dot product is zero. The calculator reports this in the Relationship box.
The cross product needs a third direction. For 2D vectors the calculator gives the z-component, which is what you get by treating them as 3D vectors with z = 0.
It is undefined, because the zero vector has no direction.
No. B × A = −(A × B), so the result points the opposite way.
It uses an isometric projection, so lengths look slightly different from their true values.
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