Let \overset{\rightarrow}{a} , \textrm{ }\textrm{ } \overset{\rightarrow}{b} \textrm{ }\textrm{ } and \textrm{ }\textrm{ } \overset{\rightarrow}{c} be three unit vectros such that \overset{\rightarrow}{a} + \overset{\rightarrow}{b} + \overset{\rightarrow}{c} = \overset{\rightarrow}{0} . \textrm{ }\textrm{ } If \textrm{ }\textrm{ } \lambda = \overset{\rightarrow}{a} . \textrm{ } \overset{\rightarrow}{b} + \overset{\rightarrow}{b} . \textrm{ }\textrm{ } \overset{\rightarrow}{c} + \overset{\rightarrow}{c} . \textrm{ }\textrm{ } \overset{\rightarrow}{a} and \overset{\rightarrow}{d} = \overset{\rightarrow}{a} \times \overset{\rightarrow}{b} + \overset{\rightarrow}{b} \times \overset{\rightarrow}{c} + \overset{\rightarrow}{c} \times \overset{\rightarrow}{a}, then the ordered pair, \left(\right. \lambda , \overset{\rightarrow}{d} \left.\right) is equal to:

Engineering

Mathematics

Vector Product of Two Vectors

Question

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

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

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

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

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

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${| \vec{a} + \vec{b} + \vec{c} |}^{2} = 0$

$3 + 2 (\vec{a} . \vec{b} + \vec{b} . \vec{c} + \vec{c} . \vec{a}) = 0$

$(\vec{a} . \vec{b} + \vec{b} . \vec{c} + \vec{c} . \vec{a}) = \frac{- 3}{2}$

$\Rightarrow λ = \frac{- 3}{2}$

$\vec{d} = \vec{a} \times \vec{b} + \vec{b} \times (- \vec{a} - \vec{b}) + (- \vec{a} - \vec{b}) \times \vec{a}$

$= \vec{a} \times \vec{b} + \vec{a} \times \vec{b} + \vec{a} \times \vec{b}$

$\vec{d} = 3 (\vec{a} \times \vec{b})$