The angle $\theta$ between two vectors $\mathbf{a}$ and $\mathbf{b}$ can be found using the dot product formula:

$\cos(\theta) = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| |\mathbf{b}|}$

Step 1: Find the dot product $\mathbf{a} \cdot \mathbf{b}$

Given: $\mathbf{a} = \langle 0, 2, -3 \rangle \quad \text{and} \quad \mathbf{b} = \langle 1, -3, 5 \rangle$

The dot product is calculated as:

$\mathbf{a} \cdot \mathbf{b} = (0)(1) + (2)(-3) + (-3)(5) = 0 - 6 - 15 = -21$

Step 2: Find the magnitudes $|\mathbf{a}|$ and $|\mathbf{b}|$

$|\mathbf{a}| = \sqrt{0^2 + 2^2 + (-3)^2} = \sqrt{0 + 4 + 9} = \sqrt{13}$

$|\mathbf{b}| = \sqrt{1^2 + (-3)^2 + 5^2} = \sqrt{1 + 9 + 25} = \sqrt{35}$

Step 3: Use the formula to find $\theta$

$\cos(\theta) = \frac{-21}{\sqrt{13} \cdot \sqrt{35}} = \frac{-21}{\sqrt{455}}$

$\theta = \cos^{-1}\left( \frac{-21}{\sqrt{455}} \right)$

Let me calculate the exact value of the angle.The angle between the vectors $\mathbf{a} = \langle 0, 2, -3 \rangle$ and $\mathbf{b} = \langle 1, -3, 5 \rangle$ is approximately 2.965 radians.

Would you like more details or have any further questions?

Here are some related questions:

1. How do you calculate the dot product between two vectors?
2. What is the physical interpretation of the angle between two vectors?
3. How would you calculate the angle between two vectors in degrees instead of radians?
4. What does it mean when the dot product of two vectors is zero?
5. How can the magnitude of a vector be interpreted geometrically?

Tip: The dot product can be useful in determining if two vectors are orthogonal, parallel, or have some general angle between them.