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

Multiplying a vector $\overrightarrow{\mathrm{A}}$ with a positive number $\lambda$ gives a vector whose magnitude is changed by the factor $\lambda$ but the direction is the same as that of $\vec{A}$.

$|\lambda \vec{A}|=\lambda|\vec{A}| \quad(\text { if } \lambda>0$

For example, if $\vec{A}$ is multiplied by 2 , the resultant vector $2 \vec{A}$ is in the same direction as $\vec{A}$ and has a magnitude twice of $|\overrightarrow{\mathrm{A}}|$ as shown in figure (a).

Multiplying a vector $\vec{A}$ by a negative number $\lambda$ gives a vector $\lambda \vec{A}$ whose direction is opposite to the direction of $\vec{A}$ and whose magnitude is $-\lambda$ times $|\vec{A}|$.

For example, multiplying a given vector $\overrightarrow{\mathrm{A}}$ by negative numbers say $-1$ and $-1.5$, gives vectors as shown in figure (b).

The factor $\lambda$ by which a vector $\overrightarrow{\mathrm{A}}$ is multiplied could be a scalar having its own physical dimension. Then, the dimension of $\lambda \overrightarrow{\mathrm{A}}$ is the product of the dimension of $\lambda$ and $\overrightarrow{\mathrm{A}}$. For example, if we multiply a constant velocity vector by duration (of time), we get a displacement vector.

Dimensions of $\vec{v}=\mathrm{m} / \mathrm{s}$

Dimensions of $\overrightarrow{v t}=\frac{\mathrm{m}}{\mathrm{s}} \cdot \mathrm{s}=\mathrm{m}$