Determine the rotation matrix for the follwing angle and direction:
theta  = 45^deg counter clockwise

To determine the rotation matrix for an angle \(\theta = 45^\circ\) in the counterclockwise direction, we use the standard 2D rotation matrix formula:

\[
R(\theta) = \begin{bmatrix}
\cos\theta & -\sin\theta \\
\sin\theta & \cos\theta
\end{bmatrix}
\]

Substituting \(\theta = 45^\circ\):

1. \(\cos(45^\circ) = \frac{\sqrt{2}}{2}\)
2. \(\sin(45^\circ) = \frac{\sqrt{2}}{2}\)

Thus, the rotation matrix \(R(45^\circ)\) is:

\[
R(45^\circ) = \begin{bmatrix}
\frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\
\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2}
\end{bmatrix}
\]

This matrix can be used to rotate a vector counterclockwise by 45 degrees.

Would you like to know more details or have any further questions?

Here are 5 related questions you might find useful:
1. How would the rotation matrix change for a clockwise rotation by 45 degrees?
2. What is the rotation matrix for an angle of 90 degrees?
3. How can we use this rotation matrix to rotate a vector in 3D space?
4. What is the effect of applying the rotation matrix twice in succession?
5. How does the determinant of a rotation matrix relate to its properties?

**Tip:** Remember that rotation matrices are orthogonal, which means their inverse is equal to their transpose.

Explore how to determine the rotation matrix for an angle of 45 degrees counterclockwise. Understand the 2D rotation matrix formula, its application, and implications. Suitable for advanced high school students and anyone interested in rotational transformations.

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