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 $\vec{u}=(x_1,y_1)$: $$\lambda\vec{u}=\lambda\cdot(x_1,y_1)=(\lambda\cdot x_1,\lambda\cdot y_1)$$
If $\vec{u}=(-1,3)$ and $\lambda=3$, then: $$\lambda\vec{u}=3\cdot (-1,3)=(-3,9)$$
Properties of the product of real numbers and a vector:
- $\lambda(\vec{u}+\vec{v})=\lambda\vec{u}+\lambda\vec{v}$
- $(\lambda+\mu)\vec{u}=\lambda\vec{u}+\mu\vec{u}$
- $\lambda(\mu\vec{u})=(\lambda\mu)\vec{u}$
- $1\cdot\vec{u}=\vec{u}$