Question

If \( \vec{a}, \vec{b} \) and \( \vec{c} \) are vectors and \( \vec{a} + \vec{b} + \vec{c} = \vec{0} \), then find the value of \( (\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) \).

Hint:

This is a standard and very common vector algebra problem. The technique of "squaring" a vector sum by taking its dot product with itself is extremely useful and should be remembered. It's the vector equivalent of squaring a scalar algebraic expression.

Answer 1

Step 1: Understanding the Concept:
We can find the value of the expression involving dot products by using the given vector sum. The key is to take the dot product of the vector sum with itself, which introduces the square of the magnitudes and the dot product terms we are interested in.
Step 2: Key Formula or Approach:
- Given: \( \vec{a} + \vec{b} + \vec{c} = \vec{0} \)
- Square this equation by taking the dot product with itself: \( (\vec{a} + \vec{b} + \vec{c}) \cdot (\vec{a} + \vec{b} + \vec{c}) = \vec{0} \cdot \vec{0} \)
- Use the property \( \vec{v} \cdot \vec{v} = |\vec{v}|^2 \).
Step 3: Detailed Explanation or Calculation:
Start with the given vector sum: \[ \vec{a} + \vec{b} + \vec{c} = \vec{0} \] Take the dot product of both sides with \( (\vec{a} + \vec{b} + \vec{c}) \): \[ (\vec{a} + \vec{b} + \vec{c}) \cdot (\vec{a} + \vec{b} + \vec{c}) = 0 \] Expand the left side: \[ \vec{a} \cdot \vec{a} + \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c} + \vec{b} \cdot \vec{a} + \vec{b} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} + \vec{c} \cdot \vec{b} + \vec{c} \cdot \vec{c} = 0 \] 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, we can solve for the expression we want: \[ 2(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) = -(|\vec{a}|^2 + |\vec{b}|^2 + |\vec{c}|^2) \] \[ \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} = -\frac{1}{2}(|\vec{a}|^2 + |\vec{b}|^2 + |\vec{c}|^2) \] Step 4: Final Answer:
The value of \( \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} \) is \( -\frac{1}{2}(|\vec{a}|^2 + |\vec{b}|^2 + |\vec{c}|^2) \).