Question 26 - 2020 (07 Jan Shift 2)

Let $\vec{a}$, $\vec{b}$ and $\vec{c}$ be three unit vectors such that $\vec{a} + \vec{b} + \vec{c} = \vec{0}$. If $\lambda = \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}$ and $\vec{d} = \vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a}$, then the ordered pair $(\lambda, \vec{d})$ is equal to:

(1) $\left(\frac{3}{2}, 3\vec{a} \times \vec{c}\right)$

(2) $\left(-\frac{3}{2}, 3\vec{c} \times \vec{b}\right)$

(3) $\left(\frac{3}{2}, 3\vec{b} \times \vec{c}\right)$

(4) $\left(-\frac{3}{2}, 3\vec{a} \times \vec{b}\right)$