Vectors: \(\mathbf{u},\) \(\mathbf{v},\) \(\mathbf{w}\)

Zero vector: \(\mathbf{0}\)

Zero vector: \(\mathbf{0}\)

Coordinate of vectors: \(X\), \(Y\), \(Z\)

Real numbers: \(\lambda\), \(\mu\)

Real numbers: \(\lambda\), \(\mu\)

- Scalar multiplication of a vector \(\mathbf{u} \ne \mathbf{0}\) by a number \(\lambda \ne 0\) is the vector \(\mathbf{w}\), whose magnitude is equal to \(\left| \lambda \right| \cdot \left| \mathbf{u} \right|\), the direction of which coincides with the vector \(\mathbf{u}\) at \(\lambda \gt 0\) and opposite to it at \(\lambda \lt 0\).

\(\mathbf{w} = \lambda \mathbf{u},\;\) \(\left| \mathbf{w} \right| = \left| \lambda \right| \cdot \left| \mathbf{u} \right|\) - The product of a vector \(\mathbf{u}\) by a number \(\lambda\) at \(\lambda = 0\) and/or \(\mathbf{u} = \mathbf{0}\) is the zero vector \(\mathbf{0}\).

Scalar multiplication has the following linear properties: - Commutativity of scalar multiplication

\(\lambda \mathbf{u} = \mathbf{u}\lambda \) - Distributivity of scalar multiplication over addition of numbers

\(\left( {\lambda + \mu } \right)\mathbf{u} =\) \( \lambda \mathbf{u} + \mu \mathbf{u}\) - Distributivity of scalar multiplication over addition of vectors

\(\lambda \left( {\mathbf{u} + \mathbf{v}} \right) =\) \(\lambda \mathbf{u} + \lambda \mathbf{v}\) - Associativity of scalar multiplication

\(\lambda \left( {\mu \mathbf{u}} \right) = \mu \left( {\lambda \mathbf{u}} \right) \) \(= \left( {\lambda \mu } \right)\mathbf{u}\) - Identity element of scalar multiplication

\(1 \cdot \mathbf{u} = \mathbf{u}\) - Scalar multiplication in coordinate form

\(\lambda \mathbf{u} = \left( {\lambda X,\lambda Y,\lambda Z} \right)\)