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

4.$\mathbf{\text{F(r)}}\mathbf{=}\mathbf{\text{r}}\mathbf{-}\mathbf{\text{ix}}\mathbf{=}\mathbf{\text{jy}}\mathbf{+}\mathbf{\text{kz}}$

A vector-valued function, generally termed as a vector function, is a mathematical function with one or more variables that has a range of multidimensional or infinite-dimensional vectors.

A functionf(r)is a linear vector function if$F(r_1+r_2)=F\left(r_1\right)+F\left(r_2\right)$and$F\left(ar\right)=aF\left(r\right)$,where a is a scalar.

The given vector function is$\text{F(r)}=\text{r}-\text{ix}=\text{jy}+\text{kz}$.

Check to see whether the given vector function is linear.

Verify that the given vector function$\text{F(r)}=\text{r}-\text{ix}=\text{jy}+\text{kz}$is linear with the help of two conditions which are defined as$\text{F}({\text{r}}_1+{\text{r}}_2)=\text{F}\left({\text{r}}_1\right)+\text{F}\left({\text{r}}_2\right)$ and$\text{F}\left(a\text{r}\right)=a\text{F}\left(\text{r}\right)$.

The function can be written as$\text{F(r)}=\text{r}-\text{(r}.\text{i)i}$

According to the first condition.

$\begin{array}{rcl}{\text{F(r}}_1+{\text{r}}_2\text{)}& =& {\text{(r}}_1+{\text{r}}_2\text{)}-\left(\right({\text{r}}_1+{\text{r}}_2).\text{i)i}\\ & =& {\text{r}}_1-({\text{r}}_1.\text{i})\text{i}+{\text{r}}_2-({\text{r}}_2.\text{i})\text{i}\\ & =& {\text{F(r}}_1\text{)}+{\text{F(r}}_2\text{)}\end{array}$

According to the second condition.

$\begin{array}{rcl}\text{F}\left(a\text{r}\right)& =& a\text{r}-\left(a\text{r.i}\right)\text{i}\\ & =& a\text{F}\left(\text{r}\right)\end{array}$

Therefore, the given vector function is a linear vector function.