Product of a real number by a vector

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

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

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

Properties of the product of real numbers and a vector:

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