1. **Problem statement:** Given vectors $\vec{a} = (-5, 4, 1)$, $\vec{b} = (3, -5, -3)$, and $\vec{c} = (5, -2, 5)$, evaluate the expression $$\left(\frac{\vec{a} \cdot \vec{b}}{\vec{b} \cdot \vec{b}}\right) \vec{b}$$ 2. **Recall the dot product formula:** $$\vec{u} \cdot \vec{v} = u_1 v_1 + u_2 v_2 + u_3 v_3$$ 3. **Calculate $\vec{a} \cdot \vec{b}$:** $$(-5)(3) + (4)(-5) + (1)(-3) = -15 - 20 - 3 = -38$$ 4. **Calculate $\vec{b} \cdot \vec{b}$:** $$3^2 + (-5)^2 + (-3)^2 = 9 + 25 + 9 = 43$$ 5. **Compute the scalar coefficient:** $$\frac{\vec{a} \cdot \vec{b}}{\vec{b} \cdot \vec{b}} = \frac{-38}{43}$$ 6. **Multiply scalar by vector $\vec{b}$:** $$\left(\frac{-38}{43}\right)(3, -5, -3) = \left(\frac{-114}{43}, \frac{190}{43}, \frac{114}{43}\right)$$ **Final answer:** $$\boxed{\left(\frac{-114}{43}, \frac{190}{43}, \frac{114}{43}\right)}$$