Question

Three vectors \( \vec{a}, \vec{b}, \vec{c} \) satisfy: \[ \vec{a} + \vec{b} + \vec{c} = \vec{0},\quad |\vec{a}| = 1,\quad |\vec{b}| = 3,\quad |\vec{c}| = 4 \] Then the value of: \[ \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} = ? \]

• A. 12
• B. -12
• C. -13
• D. 13

Hint:

When vector sums are zero, square both sides and expand using dot product properties to derive identities.

Answer 1

The Correct Option is C

We use the identity for: \[ (\vec{a} + \vec{b} + \vec{c})^2 = 0 \Rightarrow \vec{a}^2 + \vec{b}^2 + \vec{c}^2 + 2(\vec{a}\cdot\vec{b} + \vec{b}\cdot\vec{c} + \vec{c}\cdot\vec{a}) = 0 \] \[ |\vec{a}|^2 = 1,\quad |\vec{b}|^2 = 9,\quad |\vec{c}|^2 = 16 \Rightarrow 1 + 9 + 16 = 26 \] So: \[ 26 + 2(x) = 0 \Rightarrow x = \vec{a}\cdot\vec{b} + \vec{b}\cdot\vec{c} + \vec{c}\cdot\vec{a} = -13 \]