1. **Stating the problem:** We are given vectors along a path and two diagonal vectors converging to a point, with a final vertical vector labeled 5. We want to analyze the vector relationships and find the values of $x$, $y$, and $z$. 2. **Understanding the vectors:** The vectors along the horizontal path are: - $\vec{A} = 2x + y - 3z$ - $\vec{B} = -2x + 2y + z$ - $\vec{C} = 3x - y + 3z$ The two diagonal vectors converging to a point are: - $\vec{D} = x + 3z$ - $\vec{E} = 2x - y + z$ The vertical vector from the converging point is: - $\vec{F} = 5$ 3. **Vector addition rule:** The sum of vectors along a closed path or converging at a point should satisfy: $$\vec{A} + \vec{B} + \vec{C} = \vec{D} + \vec{E} + \vec{F}$$ 4. **Set up the equation:** $$ (2x + y - 3z) + (-2x + 2y + z) + (3x - y + 3z) = (x + 3z) + (2x - y + z) + 5 $$ 5. **Simplify left side:** $$ 2x + y - 3z - 2x + 2y + z + 3x - y + 3z = (2x - 2x + 3x) + (y + 2y - y) + (-3z + z + 3z) = 3x + 2y + z $$ 6. **Simplify right side:** $$ x + 3z + 2x - y + z + 5 = (x + 2x) + (-y) + (3z + z) + 5 = 3x - y + 4z + 5 $$ 7. **Equate both sides:** $$ 3x + 2y + z = 3x - y + 4z + 5 $$ 8. **Subtract $3x$ from both sides:** $$ 2y + z = - y + 4z + 5 $$ 9. **Bring all terms to one side:** $$ 2y + y + z - 4z = 5 $$ $$ 3y - 3z = 5 $$ 10. **Divide both sides by 3:** $$ y - z = \frac{5}{3} $$ **Final answer:** $$ y - z = \frac{5}{3} $$ This equation relates $y$ and $z$ based on the given vector conditions. Without additional information, we cannot find unique values for $x$, $y$, and $z$, but this relation must hold.