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Let's solve the questions step by step:

### Part (a): Find \( c \) so that the vectors are parallel.

The two vectors are:
\[
\mathbf{v_1} = \begin{pmatrix} 2 \\ c \end{pmatrix} \quad \text{and} \quad \mathbf{v_2} = \begin{pmatrix} -3 \\ 5 \end{pmatrix}.
\]

Two vectors are parallel if one is a scalar multiple of the other, i.e., there exists a scalar \( k \) such that:
\[
\mathbf{v_1} = k \mathbf{v_2}.
\]

This gives the system of equations:
\[
2 = k(-3) \quad \text{and} \quad c = k(5).
\]

From the first equation:
\[
k = \frac{2}{-3} = -\frac{2}{3}.
\]

Now, substitute \( k = -\frac{2}{3} \) into the second equation:
\[
c = -\frac{2}{3}(5) = -\frac{10}{3}.
\]

Thus, the value of \( c \) that makes the vectors parallel is:
\[
c = -\frac{10}{3}.
\]

### Part (b): Calculate the angle between the two vectors for any \( c \).

The angle \( \theta \) between two vectors \( \mathbf{v_1} \) and \( \mathbf{v_2} \) can be found using the dot product formula:
\[
\cos \theta = \frac{\mathbf{v_1} \cdot \mathbf{v_2}}{\|\mathbf{v_1}\| \|\mathbf{v_2}\|}.
\]

First, compute the dot product \( \mathbf{v_1} \cdot \mathbf{v_2} \):
\[
\mathbf{v_1} \cdot \mathbf{v_2} = 2(-3) + c(5) = -6 + 5c.
\]

Next, compute the magnitudes of \( \mathbf{v_1} \) and \( \mathbf{v_2} \):
\[
\|\mathbf{v_1}\| = \sqrt{2^2 + c^2} = \sqrt{4 + c^2},
\]
\[
\|\mathbf{v_2}\| = \sqrt{(-3)^2 + 5^2} = \sqrt{9 + 25} = \sqrt{34}.
\]

Now, substitute these into the cosine formula:
\[
\cos \theta = \frac{-6 + 5c}{\sqrt{4 + c^2} \cdot \sqrt{34}}.
\]

Thus, the cosine of the angle \( \theta \) between the vectors for any \( c \) is:
\[
\cos \theta = \frac{-6 + 5c}{\sqrt{34(4 + c^2)}}.
\]

Would you like more details on any of the steps?

Here are 5 related questions to deepen understanding:
1. How do we know that two vectors are parallel using their cross product?
2. Can vectors of different dimensions also be parallel? Why or why not?
3. How would the formula for the angle between vectors change in 3D?
4. What happens if the angle between two vectors is 90 degrees? How is that reflected in their dot product?
5. How does changing the scalar \( k \) affect the direction and magnitude of a vector?

**Tip:** The dot product helps identify the relationship between vectors—parallelism, orthogonality, or a general angle—so it's a key tool in vector analysis.

(a) Find c so that the vectors (2, c) and (-3, 5) are parallel. (b) Calculate the angle between the two vectors for any c.

Learn how to find the scalar value c so that the vectors (2, c) and (-3, 5) are parallel. Also, learn how to calculate the angle between the two vectors for any c using the dot product formula. This problem covers vector properties, parallelism, and trigonometry. Suitable for students in Grades 10-12.

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