Vector Calculator — 2D Magnitude, Dot Product, Cross Product & Angle
Enter the x and y components of two 2-D vectors to compute their magnitudes, dot product, the z-component of the cross product, the angle between them, and their vector sum — with a live diagram showing all three arrows from the origin.
Positive — vectors point in a similar direction
- 1
Ax × Bx
3 × 1 = 3 - 2
Ay × By
4 × 2 = 8 - 3
Dot product A · B
3 + 8 = 11Sum of the component products; positive means vectors point in a similar direction.
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For 2-D vectors A = (Ax, Ay) and B = (Bx, By): magnitudes |A| = √(Ax²+Ay²) and |B| = √(Bx²+By²); dot product A·B = AxBx+AyBy (scalar, encodes alignment); angle θ = arccos(A·B ÷ |A||B|); cross product z-component = AxBy−AyBx (area of parallelogram, positive = B is CCW from A); vector sum A+B = (Ax+Bx, Ay+By).
Foirmle
How this is calculated
A 2-D vector is a quantity with both magnitude and direction, represented as an ordered pair (x, y) from the origin. The magnitude (length) of vector A = (Ax, Ay) is |A| = √(Ax² + Ay²), the Euclidean distance from the origin to the point. Magnitude is always non-negative and tells you "how much" of the quantity there is, independent of direction.
The dot product A·B = Ax·Bx + Ay·By is a scalar that encodes the alignment of the two vectors. If A·B > 0, the angle between them is less than 90° (similar direction). If A·B = 0, the vectors are perpendicular (orthogonal). If A·B < 0, the angle exceeds 90° (opposing directions). Dividing by both magnitudes and taking arccos gives the exact angle θ in degrees — this is undefined when either vector has zero length.
The cross product of two 2-D vectors is not a 2-D vector: treating A and B as 3-D vectors with z = 0, their cross product A × B points purely in the z direction. The z-component is Ax·By − Ay·Bx. Its absolute value equals |A|·|B|·sin θ, which is the area of the parallelogram spanned by A and B. A positive value means B is counter-clockwise from A; negative means clockwise. The vector sum A + B is computed component-wise and is shown as the resultant arrow in the diagram.
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A·B = |A|·|B|·cos θ — it is the product of the two magnitudes scaled by the cosine of the angle between them. Equivalently, it equals |A| times the scalar projection of B onto A (or vice versa). A dot product of zero means the vectors are perpendicular; a maximum magnitude dot product means they point in exactly the same direction.
The cross product A × B is a 3-D operation that produces a vector perpendicular to both inputs. When the inputs lie in the xy-plane (z = 0), the result is purely in the z-direction: (0, 0, Ax·By − Ay·Bx). Only the z-component is meaningful in 2-D, and it is reported here as a scalar. Its absolute value gives the area of the parallelogram formed by A and B.
This calculator is for 2-D (x, y) vectors. For 3-D vectors (x, y, z), the formulas extend: |A| = √(Ax²+Ay²+Az²); dot product adds an Az·Bz term; and the cross product becomes a full 3-component vector. A dedicated 3-D vector calculator would be needed for those operations.
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TG we-Calculate Editorial Team. (2026). Vector Calculator — 2D Magnitude, Dot Product, Cross Product & Angle [Online calculator]. TG we-Calculate. https://we-calculate.com/ga/calculator/vector-calculator
TG we-Calculate Editorial Team. "Vector Calculator — 2D Magnitude, Dot Product, Cross Product & Angle." TG we-Calculate. 2026. https://we-calculate.com/ga/calculator/vector-calculator.
TG we-Calculate Editorial Team, "Vector Calculator — 2D Magnitude, Dot Product, Cross Product & Angle," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/ga/calculator/vector-calculator
@misc{wecalculate_vector_calculator, title = {Vector Calculator — 2D Magnitude, Dot Product, Cross Product & Angle}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/ga/calculator/vector-calculator}}, year = {2026}, note = {TG we-Calculate} }
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