The vector 
C
 has 
Cx = 2.00 units,
 
Cy = 3.00 units,
 and 
Cz = 3.00 units. Find the angle 
C
 makes with each of the three axes.

To find the angles a vector \(\mathbf{C}\) makes with each of the three axes (x, y, and z), we can use the following formula for the angle \(\theta\) between the vector and an axis:

\[
\cos \theta_i = \frac{C_i}{|\mathbf{C}|}
\]
where:

- \(C_i\) is the component of the vector along the \(i\)-axis (\(i\) being \(x\), \(y\), or \(z\)),
- \(|\mathbf{C}|\) is the magnitude of the vector \(\mathbf{C}\), calculated as:

\[
|\mathbf{C}| = \sqrt{C_x^2 + C_y^2 + C_z^2}
\]

Given the components \(C_x = 2.00\), \(C_y = 3.00\), and \(C_z = 3.00\), let's first calculate the magnitude of the vector:

\[
|\mathbf{C}| = \sqrt{2.00^2 + 3.00^2 + 3.00^2} = \sqrt{4.00 + 9.00 + 9.00} = \sqrt{22.00} \approx 4.69 \, \text{units}
\]

Now, let's calculate the angles with each axis:

### Angle with the x-axis:
\[
\cos \theta_x = \frac{C_x}{|\mathbf{C}|} = \frac{2.00}{4.69} \approx 0.426
\]
\[
\theta_x = \cos^{-1}(0.426) \approx 64.8^\circ
\]

### Angle with the y-axis:
\[
\cos \theta_y = \frac{C_y}{|\mathbf{C}|} = \frac{3.00}{4.69} \approx 0.640
\]
\[
\theta_y = \cos^{-1}(0.640) \approx 50.2^\circ
\]

### Angle with the z-axis:
\[
\cos \theta_z = \frac{C_z}{|\mathbf{C}|} = \frac{3.00}{4.69} \approx 0.640
\]
\[
\theta_z = \cos^{-1}(0.640) \approx 50.2^\circ
\]

Thus, the angles are:

- Angle with the x-axis: \(\theta_x \approx 64.8^\circ\)
- Angle with the y-axis: \(\theta_y \approx 50.2^\circ\)
- Angle with the z-axis: \(\theta_z \approx 50.2^\circ\)

Would you like more details or further explanation on this?

Here are 5 related questions to explore further:
1. How do we generalize the formula for the angle between two arbitrary vectors?
2. What is the geometric interpretation of the vector angle in 3D space?
3. How would these angles change if one of the components was negative?
4. Can you derive the vector dot product using these angle formulas?
5. How would you calculate the projection of vector \(\mathbf{C}\) onto each axis?

**Tip:** Always double-check your units when calculating vector magnitudes and angles to avoid confusion.

The vector C has Cx = 2.00 units, Cy = 3.00 units, and Cz = 3.00 units. Find the angle C makes with each of the three axes.

Calculate the angles a vector makes with the x, y, and z axes using its components (Cx = 2.00, Cy = 3.00, Cz = 3.00). Learn how to find the magnitude of the vector and use trigonometric formulas to determine the angles. Suitable for high school students in Grades 10-12.

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