Question

If \( |\vec{a}| = 7 \), \( |\vec{b}| = 24 \), \( |\vec{c}| = 25 \) and
\( \vec{a} + \vec{b} + \vec{c} = \vec{0} \),
find the value of \( \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} \).

Answer 1

Step 1: Given, \[ \vec{a} + \vec{b} + \vec{c} = \vec{0} \] So, \[ \vec{a} + \vec{b} = -\vec{c} \]
Step 2: Take modulus squared: \[ |\vec{a} + \vec{b}|^2 = |\vec{c}|^2 \] Step 3: Expand using dot product: \[ (\vec{a} + \vec{b}) \cdot (\vec{a} + \vec{b}) = |\vec{c}|^2 \] \[ |\vec{a}|^2 + |\vec{b}|^2 + 2\vec{a} \cdot \vec{b} = |\vec{c}|^2 \] Substitute given magnitudes:
\[ 7^2 + 24^2 + 2\vec{a} \cdot \vec{b} = 25^2 \] \[ 49 + 576 + 2\vec{a} \cdot \vec{b} = 625 \Rightarrow 2\vec{a} \cdot \vec{b} = 0 \Rightarrow \vec{a} \cdot \vec{b} = 0 \]
Step 4: Use \( \vec{c} = -(\vec{a} + \vec{b}) \)
Compute: \[ \vec{b} \cdot \vec{c} = \vec{b} \cdot (-\vec{a} - \vec{b}) = -\vec{b} \cdot \vec{a} - |\vec{b}|^2 = -0 - 576 = -576 \] \[ \vec{c} \cdot \vec{a} = (-\vec{a} - \vec{b}) \cdot \vec{a} = -|\vec{a}|^2 - \vec{b} \cdot \vec{a} = -49 - 0 = -49 \]
Final expression: \[ \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} = 0 + (-576) + (-49) = \boxed{-625} \]