# Formulas

## Analytic Geometry # Scaling Vectors

Vectors: $$\mathbf{u},$$ $$\mathbf{v},$$ $$\mathbf{w}$$
Zero vector: $$\mathbf{0}$$
Coordinate of vectors: $$X$$, $$Y$$, $$Z$$
Real numbers: $$\lambda$$, $$\mu$$
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 $$\lambda \lt 0$$.
$$\mathbf{w} = \lambda \mathbf{u},\;$$ $$\left| \mathbf{w} \right| = \left| \lambda \right| \cdot \left| \mathbf{u} \right|$$
2. 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:
3. Commutativity of scalar multiplication
$$\lambda \mathbf{u} = \mathbf{u}\lambda$$
4. Distributivity of scalar multiplication over addition of numbers
$$\left( {\lambda + \mu } \right)\mathbf{u} =$$ $$\lambda \mathbf{u} + \mu \mathbf{u}$$
5. Distributivity of scalar multiplication over addition of vectors
$$\lambda \left( {\mathbf{u} + \mathbf{v}} \right) =$$ $$\lambda \mathbf{u} + \lambda \mathbf{v}$$
6. Associativity of scalar multiplication
$$\lambda \left( {\mu \mathbf{u}} \right) = \mu \left( {\lambda \mathbf{u}} \right)$$ $$= \left( {\lambda \mu } \right)\mathbf{u}$$
7. Identity element of scalar multiplication
$$1 \cdot \mathbf{u} = \mathbf{u}$$
8. Scalar multiplication in coordinate form
$$\lambda \mathbf{u} = \left( {\lambda X,\lambda Y,\lambda Z} \right)$$