Scalar Multiplication Tool

Math Linear Algebra • Vectors and Basic Operations

Written by STEM Calculators Team Published February 21, 2026 Updated February 24, 2026

Scale a vector by a scalar \(k\) using \(k\mathbf{V}\), and (optionally) compute the unit vector \(\hat{\mathbf{V}}=\mathbf{V}/\lVert\mathbf{V}\rVert\). Negative \(k\) reverses direction.

Scale + direction visualization

Enter \(\mathbf{V}\) and \(k\). 2D: drag to pan, wheel to zoom. 3D: drag to rotate, Shift+drag to pan, wheel to zoom.

\(\mathbf{V}\) \(k\mathbf{V}\) \(\hat{\mathbf{V}}\) (unit, dashed)

Enter a vector and click “Calculate”.

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Theory

Scalar multiplication

Scalar multiplication scales a vector by a real number \(k\). For \(\mathbf{V}=(v_1,\dots,v_n)\in\mathbb{R}^n\), \[ k\mathbf{V}=(kv_1,\;kv_2,\;\dots,\;kv_n). \]

Geometrically:

If \(k>1\), the vector stretches (longer).
If \(0
If \(k=0\), the result is the zero vector \(\mathbf{0}\).
If \(k<0\), the direction reverses and the length scales by \(|k|\).

Norm (length) and scaling rule

The Euclidean norm (length) of \(\mathbf{V}\) is \[ \lVert\mathbf{V}\rVert=\sqrt{v_1^2+v_2^2+\cdots+v_n^2}. \]

Scalar multiplication interacts with the norm in a simple way: \[ \lVert k\mathbf{V}\rVert = |k|\,\lVert\mathbf{V}\rVert. \] This is a fast correctness check for your computed \(k\mathbf{V}\).

Unit vector (normalization)

A unit vector points in the same direction as \(\mathbf{V}\) but has length \(1\): \[ \hat{\mathbf{V}}=\frac{\mathbf{V}}{\lVert\mathbf{V}\rVert}. \]

Normalization is only defined when \(\lVert\mathbf{V}\rVert\neq 0\). The zero vector has no direction, so it cannot be normalized.

Applications: direction of velocity, direction of force, surface normals in graphics, and many linear algebra proofs.

University note: scalars from other fields

In basic linear algebra, scalars usually come from \(\mathbb{R}\). More generally, vector spaces can be defined over other fields (for example \(\mathbb{C}\)). The component-wise rule for scalar multiplication is the same idea, but the scalar arithmetic changes with the field.

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