To find the angle $\theta$ between the vectors $\mathbf{A} = 2\mathbf{i} + 3\mathbf{j}$ and $\mathbf{B} = \mathbf{i} - \mathbf{j}$, we can use the formula:

$\cos \theta = \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{A}| |\mathbf{B}|}$

Step 1: Calculate the dot product $\mathbf{A} \cdot \mathbf{B}$

$\mathbf{A} \cdot \mathbf{B} = (2)(1) + (3)(-1) = 2 - 3 = -1$

Step 2: Calculate the magnitudes of $\mathbf{A}$ and $\mathbf{B}$

$|\mathbf{A}| = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13}$

$|\mathbf{B}| = \sqrt{1^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2}$

Step 3: Substitute into the cosine formula

$\cos \theta = \frac{-1}{\sqrt{13} \cdot \sqrt{2}} = \frac{-1}{\sqrt{26}}$

Step 4: Find the angle $\theta$

$\theta = \cos^{-1}\left(\frac{-1}{\sqrt{26}}\right)$

Calculating this gives:

$\theta \approx 108.21^\circ$

Thus, the angle between the vectors $2\mathbf{i} + 3\mathbf{j}$ and $\mathbf{i} - \mathbf{j}$ is approximately $108.21^\circ$.

Do you want details or have any questions? Here are some related questions you might consider:

1. What is the geometric interpretation of the dot product?
2. How do you find the angle between three-dimensional vectors?
3. Can you provide an example of vectors with a right angle?
4. What other methods can be used to find angles between vectors?
5. How does the angle change if the vectors are scaled?

Tip: Always ensure your vectors are expressed in the same coordinate system before performing calculations.