1. The problem is to solve for $c$ without using vectors. 2. Since the problem statement is brief, let's assume it involves a common algebraic equation or geometric relation where $c$ is unknown. 3. For example, if the problem is to solve for $c$ in the Pythagorean theorem $a^2 + b^2 = c^2$, we use the formula: $$c = \sqrt{a^2 + b^2}$$ 4. This formula states that the length of the hypotenuse $c$ in a right triangle is the square root of the sum of the squares of the other two sides $a$ and $b$. 5. To solve for $c$, square the known side lengths $a$ and $b$, add them, and then take the square root of the result. 6. For example, if $a=3$ and $b=4$, then: $$c = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$$ 7. Therefore, $c=5$. 8. If your problem is different, please provide the full equation or context for a precise solution.