Question

If three vectors \( \vec{a} \), \( \vec{b} \) and \( \vec{c} \) satisfying the condition \( \vec{a} + \vec{b} + \vec{c} = \vec{0} \). If \( |\vec{a}| = 3 \), \( |\vec{b}| = 4 \) and \( |\vec{c}| = 2 \), then find the value of \( \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} \).

Hint:

Whenever you are given a sum of vectors equal to zero and their magnitudes, and asked to find a sum of dot products, the most direct method is to square the vector sum equation (i.e., take its dot product with itself).

Answer 1

Step 1: Understanding the Concept:
This problem uses the properties of the dot product of vectors, specifically that the dot product of a vector with itself gives the square of its magnitude. We can use the given condition \( \vec{a} + \vec{b} + \vec{c} = \vec{0} \) to find the required value.
Step 2: Key Formula or Approach:
The key identity is \( \vec{v} \cdot \vec{v} = |\vec{v}|^2 \). We will take the dot product of the vector sum \( (\vec{a} + \vec{b} + \vec{c}) \) with itself.
We start with the given equation:
\[ \vec{a} + \vec{b} + \vec{c} = \vec{0} \] Taking the dot product of both sides with \( (\vec{a} + \vec{b} + \vec{c}) \):
\[ (\vec{a} + \vec{b} + \vec{c}) \cdot (\vec{a} + \vec{b} + \vec{c}) = \vec{0} \cdot \vec{0} \] Step 3: Detailed Explanation or Calculation:
Expanding the left side of the equation:
\[ \vec{a} \cdot \vec{a} + \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c} + \vec{b} \cdot \vec{a} + \vec{b} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} + \vec{c} \cdot \vec{b} + \vec{c} \cdot \vec{c} = 0 \] Using \( \vec{v} \cdot \vec{v} = |\vec{v}|^2 \) and the commutative property of the dot product (\( \vec{u} \cdot \vec{v} = \vec{v} \cdot \vec{u} \)):
\[ |\vec{a}|^2 + |\vec{b}|^2 + |\vec{c}|^2 + 2(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) = 0 \] Now, substitute the given magnitudes: \( |\vec{a}| = 3 \), \( |\vec{b}| = 4 \), and \( |\vec{c}| = 2 \).
\[ (3)^2 + (4)^2 + (2)^2 + 2(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) = 0 \] \[ 9 + 16 + 4 + 2(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) = 0 \] \[ 29 + 2(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) = 0 \] Now, solve for the required expression:
\[ 2(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) = -29 \] \[ \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} = -\frac{29}{2} \] Step 4: Final Answer:
The value of \( \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} \) is \( -\frac{29}{2} \).