Scaling Vectors

1. 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 \(\lambda \lt 0\).
   \(\mathbf{w} = \lambda \mathbf{u},\;\) \(\left| \mathbf{w} \right| = \left| \lambda \right| \cdot \left| \mathbf{u} \right|\)

1. The product of a vector \(\mathbf{u}\) by a number \(\lambda\) at \(\lambda = 0\) and/or \(\mathbf{u} = \mathbf{0}\) is the zero vector \(\mathbf{0}\).
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   Scalar multiplication has the following linear properties:
2. Commutativity of scalar multiplication
   \(\lambda \mathbf{u} = \mathbf{u}\lambda \)
3. Distributivity of scalar multiplication over addition of numbers
   \(\left( {\lambda + \mu } \right)\mathbf{u} =\) \( \lambda \mathbf{u} + \mu \mathbf{u}\)
4. Distributivity of scalar multiplication over addition of vectors
   \(\lambda \left( {\mathbf{u} + \mathbf{v}} \right) =\) \(\lambda \mathbf{u} + \lambda \mathbf{v}\)
5. Associativity of scalar multiplication
   \(\lambda \left( {\mu \mathbf{u}} \right) = \mu \left( {\lambda \mathbf{u}} \right) \) \(= \left( {\lambda \mu } \right)\mathbf{u}\)
6. Identity element of scalar multiplication
   \(1 \cdot \mathbf{u} = \mathbf{u}\)
7. Scalar multiplication in coordinate form
   \(\lambda \mathbf{u} = \left( {\lambda X,\lambda Y,\lambda Z} \right)\)