Answer
$\frac{4}{5}\mathbf{i}-\frac{3}{5}\mathbf{j}$
Work Step by Step
Unit vector:
For any vector $\mathbf{v}$, $\frac{\mathbf{v}}{\left\| \mathbf{v} \right\|}$ is the unit vector in the same direction as the vector $\mathbf{v}$.
Magnitude of vector is given by:
The magnitude of $\mathbf{v}=a\mathbf{i}+b\mathbf{j}$ is given by $\left\| \mathbf{v} \right\|=\sqrt{{{a}^{2}}+{{b}^{2}}}$.
Scalar multiplication of a vector is given by:
For any vector $\mathbf{v}=a\mathbf{i}+b\mathbf{j}$ and $k$ is a real number, $k\mathbf{v}=\left( ka \right)\mathbf{i}+\left( kb \right)\mathbf{j}$.
Here, $\mathbf{v}=8\mathbf{i}-6\mathbf{j}$.
So,
$\begin{align}
& \left\| \mathbf{v} \right\|=\sqrt{{{8}^{2}}+{{\left( -6 \right)}^{2}}} \\
& =\sqrt{64+36} \\
& =\sqrt{100} \\
& =10
\end{align}$
The unit vector in the direction as the vector $\mathbf{v}$ is calculated as below:
$\begin{align}
& \frac{\mathbf{v}}{\left\| \mathbf{v} \right\|}=\frac{8\mathbf{i}-6\mathbf{j}}{10} \\
& =\frac{8}{10}\mathbf{i}-\frac{6}{10}\mathbf{j} \\
& =\frac{4}{5}\mathbf{i}-\frac{3}{5}\mathbf{j}
\end{align}$
Hence, the unit vector in the direction as the vector $\mathbf{v}$ is $\frac{4}{5}\mathbf{i}-\frac{3}{5}\mathbf{j}$.
