First slide

Cross product of vectors or vector product of vectors

Question

If $\vec{a} + 2 \vec{b} + 3 \vec{c} = \vec{0}, then \vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a} =$

Moderate

A

$2 (\vec{a} \times \vec{b})$
B

$6 (\vec{b} \times \vec{c})$
C

$3 (\vec{c} \times \vec{a})$
D

$\vec{0}$

Solution

$We know that \vec{a} + \vec{b} + \vec{c} = \vec{0}, then \vec{a} \times \vec{b} = \vec{b} \times \vec{c} = \vec{c} \times \vec{a}$
$Given \vec{a} + 2 \vec{b} + 3 \vec{c} = \vec{0}$
$or \vec{a} \times 2 \vec{b} = 2 \vec{b} \times 3 \vec{c} = 3 \vec{c} \times \vec{a}$
$hence \vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a}$
$= 2 (\vec{a} \times \vec{b}) or 6 (\vec{b} \times \vec{c}) or 3 (\vec{c} \times \vec{a})$

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