1. **Problem statement:** Given three vectors \(\vec{a}, \vec{b}, \vec{c}\) such that \(|\vec{a}|=2\), \(|\vec{b}|=3\), \(|\vec{c}|=4\), and \(\vec{a} + \vec{b} + \vec{c} = \vec{0}\), find the value of \(4 \vec{a} \cdot \vec{b} + 3 \vec{b} \cdot \vec{c} + 3 \vec{c} \cdot \vec{a}\). 2. **Key formula and rules:** - The dot product is distributive and commutative: \(\vec{u} \cdot \vec{v} = \vec{v} \cdot \vec{u}\). - From \(\vec{a} + \vec{b} + \vec{c} = \vec{0}\), we have \(\vec{c} = - (\vec{a} + \vec{b})\). - The magnitude squared of a vector \(\vec{v}\) is \(|\vec{v}|^2 = \vec{v} \cdot \vec{v}\). 3. **Step 1: Express \(\vec{c}\) in terms of \(\vec{a}\) and \(\vec{b}\):** $$\vec{c} = - (\vec{a} + \vec{b})$$ 4. **Step 2: Use the magnitude of \(\vec{c}\):** $$|\vec{c}|^2 = \vec{c} \cdot \vec{c} = 4^2 = 16$$ Substitute \(\vec{c} = - (\vec{a} + \vec{b})\): $$\vec{c} \cdot \vec{c} = (-(\vec{a} + \vec{b})) \cdot (-(\vec{a} + \vec{b})) = (\vec{a} + \vec{b}) \cdot (\vec{a} + \vec{b})$$ 5. **Step 3: Expand the dot product:** $$ (\vec{a} + \vec{b}) \cdot (\vec{a} + \vec{b}) = \vec{a} \cdot \vec{a} + 2 \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{b} $$ 6. **Step 4: Substitute known magnitudes:** $$ |\vec{a}|^2 = 2^2 = 4, \quad |\vec{b}|^2 = 3^2 = 9 $$ So, $$ 16 = 4 + 2 \vec{a} \cdot \vec{b} + 9 $$ 7. **Step 5: Solve for \(\vec{a} \cdot \vec{b}\):** $$ 16 = 13 + 2 \vec{a} \cdot \vec{b} \implies 2 \vec{a} \cdot \vec{b} = 3 \implies \vec{a} \cdot \vec{b} = \frac{3}{2} $$ 8. **Step 6: Express \(\vec{b} \cdot \vec{c}\) and \(\vec{c} \cdot \vec{a}\) in terms of \(\vec{a}\) and \(\vec{b}\):** $$ \vec{b} \cdot \vec{c} = \vec{b} \cdot (-(\vec{a} + \vec{b})) = - \vec{b} \cdot \vec{a} - \vec{b} \cdot \vec{b} = - \vec{a} \cdot \vec{b} - 9 $$ $$ \vec{c} \cdot \vec{a} = (-(\vec{a} + \vec{b})) \cdot \vec{a} = - \vec{a} \cdot \vec{a} - \vec{b} \cdot \vec{a} = -4 - \vec{a} \cdot \vec{b} $$ 9. **Step 7: Substitute all into the expression:** $$ 4 \vec{a} \cdot \vec{b} + 3 \vec{b} \cdot \vec{c} + 3 \vec{c} \cdot \vec{a} = 4 \times \frac{3}{2} + 3(- \frac{3}{2} - 9) + 3(-4 - \frac{3}{2}) $$ 10. **Step 8: Simplify step-by-step:** $$ 4 \times \frac{3}{2} = 6 $$ $$ 3(- \frac{3}{2} - 9) = 3(- \frac{3}{2}) + 3(-9) = - \frac{9}{2} - 27 = -4.5 - 27 = -31.5 $$ $$ 3(-4 - \frac{3}{2}) = 3(-4) + 3(- \frac{3}{2}) = -12 - \frac{9}{2} = -12 - 4.5 = -16.5 $$ Sum all: $$ 6 - 31.5 - 16.5 = 6 - 48 = -42 $$ **Final answer:** $$4 \vec{a} \cdot \vec{b} + 3 \vec{b} \cdot \vec{c} + 3 \vec{c} \cdot \vec{a} = -42$$