Product of a real number by a vector

The product of a real number $$\lambda$$ per a vector $$\vec{u}$$ is another vector $$\lambda\vec{u}$$ that has:

• The same angle as $$\vec{u}$$.
• Its magnitude is equal to that of $$\vec{u}$$ times the absolute value of $$\lambda$$. $$$ |\lambda\vec{u}|=|\lambda|\cdot|\vec{u}|$$$
• It has the same direction as $$\vec{u}$$ if $$\lambda>0$$ and the opposite one if $$\lambda<0$$. From this we can deduce that if $$\lambda=0$$ or if $$\vec{u}=\vec{0}$$, then $$\lambda\vec{u}=\vec{0}$$.

To obtain the components of the vector $$\lambda\vec{u}$$ it is enough to multiply by $$\lambda$$ the components of $$\vec{u}$$. If $$\vec{u}=(x_1,y_1)$$: $$$ \lambda\vec{u}=\lambda\cdot(x_1,y_1)=(\lambda\cdot x_1,\lambda\cdot y_1)$$$

Properties of the product of real numbers and a vector:

1. $$\lambda(\vec{u}+\vec{v})=\lambda\vec{u}+\lambda\vec{v}$$
2. $$(\lambda+\mu)\vec{u}=\lambda\vec{u}+\mu\vec{u}$$
3. $$\lambda(\mu\vec{u})=(\lambda\mu)\vec{u}$$
4. $$1\cdot\vec{u}=\vec{u}$$