vector question 3

**daphnewoon** · October 24th 2009, 09:27 AM

Show that if vectors a, b, c and d lie in the same plane, then (a´ b)´ (c ´ d) = 0.

**Soroban** · October 24th 2009, 06:39 PM

Hello, daphnewoon!

I have an "eyeball" proof . . .

Show that if vectors $\vec{a}, \vec{b}, \vec{c},\vec{d}$ lie in the same plane,

. . then: . $(\vec{a} \times \vec{b}) \times (\vec{c} \times \vec{d}) \:=\:\vec{0}$

$\text{Since }\vec{a}\text{ and }\vec{b}\text{ are in the same plane, }\vec{m} \:=\:\vec{a}\times\vec{b}\:\text{ is normal to the plane.}$

$\text{Since }\vec{c}\text{ and }\vec{d}\text{ are in the same plane, }\vec{n} \:=\:\vec{c} \times\vec{d}\:\text{ is normal to the plane.}$

$\text{Since }\vec{a},\vec{b},\vec{c},\vec{d}\text{ are in the same plane: }\:\vec{m} \parallel \vec{n}$

$\text{We have: }\;(\vec{a} \times \vec{b}) \times (\vec{c} \times \vec{d}) \;=\;\vec{m}\times\vec{n},\;\text{ where }\vec{m} \parallel \vec{n}$

. . $\text{and the cross-product of two parallel vectors is }\vec{0}.$

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