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To determine the angle between the vectors \( \mathbf{a} = (2, 3) \) and \( \mathbf{b} = (-1, 5) \) using the dot product, we can follow these steps:

### Formula
The formula for finding the angle \( \theta \) between two vectors using the dot product is:

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

Where:
- \( \mathbf{a} \cdot \mathbf{b} \) is the dot product of the vectors.
- \( |\mathbf{a}| \) and \( |\mathbf{b}| \) are the magnitudes (lengths) of the vectors.

### Step 1: Calculate the dot product
The dot product \( \mathbf{a} \cdot \mathbf{b} \) for vectors \( \mathbf{a} = (2, 3) \) and \( \mathbf{b} = (-1, 5) \) is calculated as:

\[
\mathbf{a} \cdot \mathbf{b} = (2)(-1) + (3)(5) = -2 + 15 = 13
\]

### Step 2: Calculate the magnitudes of the vectors
The magnitude of a vector \( \mathbf{v} = (x, y) \) is:

\[
|\mathbf{v}| = \sqrt{x^2 + y^2}
\]

So, for \( \mathbf{a} = (2, 3) \):

\[
|\mathbf{a}| = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13}
\]

For \( \mathbf{b} = (-1, 5) \):

\[
|\mathbf{b}| = \sqrt{(-1)^2 + 5^2} = \sqrt{1 + 25} = \sqrt{26}
\]

### Step 3: Calculate the cosine of the angle
Now we can plug these values into the formula for \( \cos(\theta) \):

\[
\cos(\theta) = \frac{13}{\sqrt{13} \cdot \sqrt{26}} = \frac{13}{\sqrt{338}}
\]

### Step 4: Solve for the angle \( \theta \)

\[
\theta = \cos^{-1} \left( \frac{13}{\sqrt{338}} \right)
\]

I will calculate this now.The angle between the vectors \( (2, 3) \) and \( (-1, 5) \), to the nearest tenth, is \( 45.0^\circ \). 

Thus, the correct answer is **C) 45.0°**.

Would you like further details or have any other questions?

Here are some related questions you might be interested in:
1. How is the dot product of vectors related to projection?
2. What is the difference between the dot product and the cross product?
3. How can you determine if two vectors are orthogonal using the dot product?
4. How do you calculate the magnitude of a 3D vector?
5. What is the geometric interpretation of the dot product?

**Tip**: The dot product can also help determine if vectors are perpendicular—if the dot product is zero, the vectors are orthogonal.

Use a dot product to determine, to the nearest tenth, the angle between vectors (2,3) and (-1,5).

Determine the angle between vectors (2,3) and (-1,5) using the dot product. This problem covers key vector operations and trigonometric concepts, such as dot product and magnitude calculation, ideal for high school students in Grades 10-12.

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