1. **Problem statement:** Determine the magnitude of the resultant vector given that $|\vec{a}| = 6$ and $|\vec{b}| = 8$. 2. **Understanding the problem:** The problem involves finding the magnitude of the resultant vector formed by vectors $\vec{a}$ and $\vec{b}$. The magnitude of a vector $\vec{v}$ is denoted $|\vec{v}|$. 3. **Formula used:** If vectors $\vec{a}$ and $\vec{b}$ are combined, the magnitude of the resultant vector $\vec{r} = \vec{a} + \vec{b}$ depends on the angle $\theta$ between them: $$ |\vec{r}| = \sqrt{|\vec{a}|^2 + |\vec{b}|^2 + 2|\vec{a}||\vec{b}|\cos\theta} $$ 4. **Important rule:** Without the angle $\theta$, we cannot find a unique magnitude. If $\vec{a}$ and $\vec{b}$ are perpendicular, $\cos\theta = 0$; if they are in the same direction, $\cos\theta = 1$; if opposite, $\cos\theta = -1$. 5. **Assuming vectors are perpendicular (common case):** $$ |\vec{r}| = \sqrt{6^2 + 8^2 + 2 \times 6 \times 8 \times 0} = \sqrt{36 + 64} = \sqrt{100} = 10 $$ 6. **Result:** The magnitude of the resultant vector is $10$ if $\vec{a}$ and $\vec{b}$ are perpendicular. If the angle is different, please provide it for an exact answer.