The problem asks to find the scalar (or dot) product of two vectors $\vec{a}$ and $\vec{b}$, where:

$\vec{a} = (-3, 4, -3), \quad \vec{b} = (-3, -1, 1)$

The scalar product (or dot product) of two vectors $\vec{a} = (a_1, a_2, a_3)$ and $\vec{b} = (b_1, b_2, b_3)$ is calculated using the formula:

$\vec{a} \cdot \vec{b} = a_1 \cdot b_1 + a_2 \cdot b_2 + a_3 \cdot b_3$

Substituting the given values:

$\vec{a} \cdot \vec{b} = (-3) \cdot (-3) + 4 \cdot (-1) + (-3) \cdot 1$

Calculating each term:

$\vec{a} \cdot \vec{b} = 9 - 4 - 3 = 2$

Thus, the scalar product $\vec{a} \cdot \vec{b}$ is $2$.

Would you like further details or have any questions? Here are some related questions you might find interesting:

1. What are the geometric interpretations of the dot product?
2. How can the dot product help calculate the angle between two vectors?
3. What are the properties of the dot product?
4. How is the cross product of vectors different from the dot product?
5. Can you calculate the dot product in higher-dimensional spaces?

Tip: Remember that the dot product is zero when vectors are perpendicular.