To find the values of $x$ and $y$ for a vector with magnitude $5$ units and direction $\theta = \tan^{-1} \left( \frac{3}{4} \right)$, we can use the following steps:

1. Determine the angle $\theta$: $\theta = \tan^{-1} \left( \frac{3}{4} \right)$

2. Find $\cos \theta$ and $\sin \theta$:

   Since $\theta = \tan^{-1} \left( \frac{3}{4} \right)$, we know: $\tan \theta = \frac{3}{4}$

   From trigonometric identities and the definition of tangent: $\tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{3}{4}$

   We can form a right triangle with opposite side $3$, adjacent side $4$, and hypotenuse $5$ (from the Pythagorean theorem).

   Thus: $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{4}{5}$ $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{3}{5}$

3. Use the magnitude and direction to find $x$ and $y$: The vector ( \