Explain the meaning of multiplication of vectors by real numbers with an example.

(N/A) Multiplying a vector $\vec{A}$ with a positive real number $\lambda$ results in a new vector $\lambda \vec{A}$ whose magnitude is $\lambda$ times the magnitude of $\vec{A}$,and its direction remains the same as that of $\vec{A}$.
$|\lambda \vec{A}| = \lambda |\vec{A}|$ (for $\lambda > 0$)
For example,if $\vec{A}$ is multiplied by $2$,the resultant vector $2\vec{A}$ points in the same direction as $\vec{A}$ and has a magnitude twice that of $|\vec{A}|$,as shown in figure $(a)$.
Multiplying a vector $\vec{A}$ by a negative real number $\lambda$ results in a vector $\lambda \vec{A}$ whose direction is opposite to that of $\vec{A}$ and whose magnitude is $|\lambda|$ times the magnitude of $\vec{A}$.
For example,multiplying $\vec{A}$ by $-1$ and $-1.5$ gives vectors as shown in figure $(b)$.
The factor $\lambda$ can also be a scalar with physical dimensions. In such cases,the dimension of the resulting vector $\lambda \vec{A}$ is the product of the dimensions of $\lambda$ and $\vec{A}$. For instance,multiplying a constant velocity vector $\vec{v}$ by a time interval $t$ gives a displacement vector $\vec{d} = \vec{v}t$.
Dimensions of $\vec{v} = [LT^{-1}] = m/s$
Dimensions of $\vec{v}t = [LT^{-1}] \cdot [T] = [L] = m$