Are the following linear vector functions? Prove your conclusions using (7.2).

5. $\mathbf{F}\mathbf{\left(}\mathbf{r}\mathbf{\right)}\mathbf{=}\mathbf{A}\mathbf{\times }\mathbf{r}$where is a given vector.

A vector-valued function generally called a vector function, is a mathematical function of one or more variables that has a series of multidimensional or infinite-dimensional vectors.

A function of a vector, sayf (r), is called linear if,$\mathbf{f}\mathbf{(}{\mathbf{r}}_{\mathbf{1}}\mathbf{+}{\mathbf{r}}_{\mathbf{2}}\mathbf{)}\mathbf{=}\mathbf{f}\mathbf{\left(}{\mathbf{r}}_{\mathbf{1}}\mathbf{\right)}\mathbf{+}\mathbf{f}\mathbf{\left(}{\mathbf{r}}_{\mathbf{2}}\mathbf{\right)}$, and f (ar) = af (r), whereis a scalar.

For example, F (r) = br (where b is a scalar) is a linear vector function of r.

The given vector function is

$\mathrm{F}\left(\mathrm{r}\right)=\mathrm{A}\times \mathrm{r}$

Check to see whether the given vector function is linear.

Verify that the given vector function is linear with the help of two conditions which are defined as below.

$\mathrm{F}({\mathrm{r}}_1+{\mathrm{r}}_2)=\mathrm{F}\left({\mathrm{r}}_1\right)+\mathrm{F}\left({\mathrm{r}}_2\right)$ ….. (1)

$\mathrm{F}\left(\mathrm{ar}\right)=\mathrm{aF}\left(\mathrm{r}\right)$ ….. (2)

Apply the first condition.

$$\begin{gathered} \mathrm{F}({\mathrm{r}}_1+{\mathrm{r}}_2)=A\times ({\mathrm{r}}_1+{\mathrm{r}}_2) \\ =(\mathrm{A}\times {\mathrm{r}}_1)+(\mathrm{A}\times {\mathrm{r}}_2) \\ =\mathrm{F}\left({\mathrm{r}}_1\right)+\mathrm{F}\left({\mathrm{r}}_2\right) \end{gathered}$$

Apply the second condition.

$$\begin{gathered} \mathrm{F}\left(\mathrm{ar}\right)=\mathrm{A}\times \mathrm{ar} \\ =\mathrm{aF}\left(\mathrm{r}\right) \end{gathered}$$

Therefore, as the equations (1) and (2) solved the given vector function is linear.