Vector Magnitude 2D⚠ unverified

Understanding the Parameters

• a_x (m) — Length.
• a_y (m) — Length.
• Output |a| (m) — Magnitude.

How to Calculate

1. Enter Length as ax (default 3 m). Use the unit menu when you need a different unit.
2. Enter Length as ay (default 4 m). Use the unit menu when you need a different unit.
3. Click Calculate. The card evaluates $|a|=\sqrt{a_x^2+a_y^2}$ and shows the result in the declared output unit; Result in (when present) converts that number.

The Science: A Rigorous Derivation (Approaching a Proof)

A Euclidean vector is an arrow: magnitude and direction. The dot product $\mathbf{a}\cdot\mathbf{b} = |a||b|\cos\theta$ is the unique symmetric bilinear form inducing the Euclidean norm. The cross product in $\mathbb{R}^3$ is the unique alternating bilinear map (up to scale) whose magnitude is the parallelogram area and whose direction follows the right-hand rule. The scalar triple product $\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})$ is the signed volume of the parallelepiped, and vanishes iff the three vectors are coplanar.

Dimensional check. The declared output unit is m; ToolBox evaluates the formula in SI (radians internally for every trigonometric call) and python-calc converts to the unit on the card.

History and Development

• Gibbs (1881 notes, 1901 textbook) and Heaviside. The modern $\mathbf{a}\cdot\mathbf{b}$ and $\mathbf{a}\times\mathbf{b}$ notation, distilled from Hamilton's quaternions.

• Grassmann, Ausdehnungslehre (1844). Exterior algebra — the cross product is a $2$-vector in $3$ dimensions identified with a vector via the Hodge star.

• Hamilton, quaternions (1843). $ij=k$ is the cross product plus a scalar (the negative of the dot product). Gibbs/Heaviside split that into two operations engineers actually use.

Related Concepts: Vector Magnitude 3D, Dot Product, Angle Between Vectors, Cross Product Magnitude, Scalar Triple Product, Scalar Projection, Direction Cosine