Mathematics

Problem

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 : [JEE-MAIN ONLINE-2020] (1)

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

(3)

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

(2)

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

(4)

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