First slide

Cross product of vectors or vector product of vectors

Question

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

Easy

A

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

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

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

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

Solution

${|\vec{a} + \vec{b} + \vec{c}|}^{2} = 0$
$3 + 2 (\vec{a} . \vec{b} + \vec{b} . \vec{c} + \vec{c} . \vec{a}) = 0$
$(\vec{a} . \vec{b} + \vec{b} . \vec{c} + \vec{c} . \vec{a}) = \frac{- 3}{2}$
$\Rightarrow λ = \frac{- 3}{2}$
$\vec{d} = \vec{a} \times \vec{b} + \vec{b} \times (- \vec{a} - \vec{b}) + (- \vec{a} - \vec{b}) \times \vec{a}$
$= \vec{a} \times \vec{b} + \vec{a} \times \vec{b} + \vec{a} \times \vec{b}$
$\vec{d} = 3 (\vec{a} \times \vec{b})$

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