Step 1: Conceptualization:
The problem involves a relationship between three vectors and their magnitudes. The objective is to determine the angle between two specific vectors, $\vec{a}$ and $\vec{b}$, utilizing the dot product formula. This formula connects the dot product of two vectors to their magnitudes and the cosine of the angle between them.
Step 2: Core Principle:
The angle $\theta$ between vectors $\vec{a}$ and $\vec{b}$ is defined by the equation $\cos\theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}$. The value of $\vec{a} \cdot \vec{b}$ can be derived by algebraic manipulation of the given vector equation.
Step 3: Derivation Process:
Given the equation $\vec{a} + \vec{b} + \vec{c} = \vec{0}$.
To find the angle between $\vec{a}$ and $\vec{b}$, isolate $\vec{c}$:
\[ \vec{a} + \vec{b} = -\vec{c} \]Taking the dot product of both sides with themselves is equivalent to squaring the magnitude of each side:
\[ (\vec{a} + \vec{b}) \cdot (\vec{a} + \vec{b}) = (-\vec{c}) \cdot (-\vec{c}) \]\[ |\vec{a} + \vec{b}|^2 = |-\vec{c}|^2 = |\vec{c}|^2 \]Expanding the left side yields:
\[ \vec{a} \cdot \vec{a} + 2(\vec{a} \cdot \vec{b}) + \vec{b} \cdot \vec{b} = |\vec{c}|^2 \]\[ |\vec{a}|^2 + 2(\vec{a} \cdot \vec{b}) + |\vec{b}|^2 = |\vec{c}|^2 \]Substitute the given magnitudes: $|\vec{a}| = 3, |\vec{b}| = 5, |\vec{c}| = 7$.
\[ 3^2 + 2(\vec{a} \cdot \vec{b}) + 5^2 = 7^2 \]\[ 9 + 2(\vec{a} \cdot \vec{b}) + 25 = 49 \]\[ 34 + 2(\vec{a} \cdot \vec{b}) = 49 \]Solve for the dot product:
\[ 2(\vec{a} \cdot \vec{b}) = 49 - 34 = 15 \]\[ \vec{a} \cdot \vec{b} = \frac{15}{2} \]Now, calculate the angle $\theta$ between $\vec{a}$ and $\vec{b}$:
\[ \cos\theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|} = \frac{15/2}{3 \times 5} = \frac{15/2}{15} = \frac{1}{2} \]For $\theta$ in the range $[0, \pi]$, the angle satisfying $\cos\theta = \frac{1}{2}$ is $\theta = \frac{\pi}{3}$.
Step 4: Conclusion:
The angle between $\vec{a}$ and $\vec{b}$ is $\frac{\pi}{3}$.