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Question Number 78652 by peter frank last updated on 19/Jan/20

Answered by mr W last updated on 08/Feb/20

$$threevectorsarenon-coplanarif$$$$andonlyiftheirscalartripleproduct$$$$isnotequal0.$$$$$$$$(\mathit{a}+2\mathit{b}+3\mathit{c})\cdot \left[(\lambda \mathit{b}+4\mathit{c})\times (2\lambda -1)\mathit{c}\right]$$$$=(2\lambda -1)(\mathit{a}+2\mathit{b}+3\mathit{c})\cdot (\lambda \mathit{b}\times \mathit{c}+4\mathit{c}\times \mathit{c})$$$$=(2\lambda -1)[\lambda \mathit{a}\cdot \mathit{b}\times \mathit{c}+4\mathit{a}\cdot \mathit{c}\times \mathit{c}+2\lambda \mathit{b}\cdot \mathit{b}\times \mathit{c}+8\mathit{b}\cdot \mathit{c}\times \mathit{c}+3\lambda \mathit{c}\cdot \mathit{b}\times \mathit{c}+12\mathit{c}\cdot \mathit{c}\times \mathit{c}]$$$$=(2\lambda -1)\lambda (\mathit{a}\cdot \mathit{b}\times \mathit{c})\ne 0$$$$since\mathit{a},\mathit{b},\mathit{c}arenon-coplanar,i.e.\mathit{a}\cdot \mathit{b}\times \mathit{c}\ne 0$$$$(2\lambda -1)\lambda \ne 0$$$$\Rightarrow \lambda \ne 0and\lambda \ne \frac{1}{2}$$$$\Rightarrow answer\left(c\right)iscorrect.$$

Commented by peter frank last updated on 08/Feb/20

$$thankyouverymuch$$

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