91Ó°ÊÓ

The vector is provided.

The magnitude of a vector $\stackrel{→}{r}=x\hat{i}+y\hat{j}+z\hat{k}$is

$\left|\stackrel{→}{r}\right|\mathbf{=}\sqrt{{\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}\mathbf{+}{\mathbf{z}}^{\mathbf{2}}}.....\left(\mathrm{l}\right)$

The unit vector along a vector $\stackrel{→}{r}$is

$\hat{\mathbf{r}}\mathbf{=}\frac{\stackrel{\mathbf{→}}{\mathbf{r}}}{\left|\stackrel{→}{r}\right|}\mathbf{}\mathbf{}.....\left(\mathrm{ll}\right)$

The vector $\stackrel{→}{g}$ in coordinate form is

$\stackrel{→}{g}=\left(4\hat{i}+7\hat{j}\right)\mathrm{m}$

The graphical representation of the vector is as follows

From equation (I) the magnitude of vector $\stackrel{→}{g}$is

$$\begin{gathered} \left|\stackrel{→}{g}\right|=\sqrt{4^2+7^2}\mathrm{m} \\ =8.06\mathrm{m} \end{gathered}$$

Thus, the required magnitude is $8.06\mathrm{m}$.

From equation (II) the unit vector along $\stackrel{→}{g}$is

$$\begin{gathered} \hat{g}=\frac{\left(4\hat{\mathrm{i}}+7\hat{\mathrm{j}}\right)\mathrm{m}}{8.06\mathrm{m}} \\ =0.496\hat{i}+0.868\hat{j} \end{gathered}$$

Thus, the required unit vector is$0.496\hat{i}+0.868\hat{j}$

The graphical representation of $\hat{g}$ with the tail at is shown in the image in Step 3.

From equation (II), the product of the magnitude and the unit vector will just return the original vector, that is

$$\begin{gathered} \left|\stackrel{→}{g}\right|\hat{g}=\stackrel{→}{g} \\ =\left(4\hat{i}+7\hat{j}\right)m \end{gathered}$$

Thus, the required product is$\left(4\hat{i}+7\hat{j}\right)m$