If c=3 a-2 b prove that [a b c]=0 - Vidyadip

Question

If $\bar{c}=3 \bar{a}-2 \bar{b}$ prove that $[\bar{a} \bar{b} \bar{c}]=0$

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Answer

We use the results $: \bar{b} \times \bar{b}=0$ and if in a scalar triple product, two vectors are equal, then

the scalar triple product is zero.

$\begin{aligned} {\left[\begin{array}{lll}\bar{a} & \bar{b} & \bar{c}\end{array}\right] } & =\bar{a} \cdot(\bar{b} \times \bar{c}) \\ & =\bar{a} \cdot[\bar{b} \times(\bar{a}-2 \bar{b})] \\ & =\bar{a} \cdot(3 \bar{b} \times \bar{a}-2 \bar{b} \times \bar{b}) \\ & =\bar{a} \cdot(3 \bar{b} \times \bar{a}-\overline{0}) \\ & =3 \bar{a} \cdot(\bar{b} \times \bar{a})=3 \times 0=0\end{aligned}$

Alternative Method:

$\begin{aligned} & \bar{c}=3 \bar{a}-2 \bar{b} \\ & \therefore \bar{c} \text { is a linear combination of } \bar{a} \text { and } \bar{b} \\ & \therefore \bar{a}, \bar{b}, \bar{c} \text { are coplanar } \\ & \therefore[\bar{a} \bar{b} \bar{c}]=0 .\end{aligned}$

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