To find the magnitude and direction of the vector $(4, 4)$, we will utilize Pythagoras’ theorem and some basic principles of trigonometry.

Step 1: Calculate the Magnitude of the Vector

The magnitude of a vector $(x, y)$ is calculated using the formula:

For the vector $(4, 4)$, we substitute $x = 4$ and $y = 4$ into the formula:

Thus, the magnitude of the vector $(4, 4)$ is $4\sqrt{2}$.

Step 2: Determine the Direction of the Vector

The direction of a vector is typically expressed as the angle $\theta$ that it forms with the positive x-axis. We can find this angle using the tangent function, which relates the angle to the ratio of the y-component to the x-component of the vector:

For the vector $(4, 4)$, this relationship is:

To find $\theta$, we take the inverse tangent (or arctan) of $1$:

In degrees, $\tan^{-1}(1)$ equals $45^\circ$. Therefore, the direction of the vector $(4, 4)$ is $45^\circ$ from the positive x-axis.

Summary

In conclusion, the magnitude of the vector $(4, 4)$ is $4\sqrt{2}$, and its direction is $45^\circ$ relative to the positive x-axis.