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Question Number 141129 by bobhans last updated on 16/May/21

$$Forwhatvaluesof\lambda arethe$$$$vectors\lambda \hat{i}+2\hat{j}+\hat{k},3\hat{i}+4\hat{j}+\lambda \hat{k}$$$$,\hat{j}+\hat{k}coplanar$$

Answered by EDWIN88 last updated on 16/May/21

$$\left(\begin{array}{c}3\\ 4\\ \lambda \end{array}\right)=\mathrm{m}\left(\begin{array}{c}\lambda \\ 2\\ 1\end{array}\right)+\mathrm{n}\left(\begin{array}{c}0\\ 1\\ 1\end{array}\right)$$$$\Rightarrow \mathrm{m}\lambda =3;\lambda =\frac{3}{\mathrm{m}}$$$$\Rightarrow \{\begin{array}{l}2\mathrm{m}+\mathrm{n}=4\\ \mathrm{m}+\mathrm{n}=\lambda \end{array}\Rightarrow \mathrm{m}=4-\lambda$$$$\Rightarrow \mathrm{m}=4-\frac{3}{\mathrm{m}};{\mathrm{m}}^2-4\mathrm{m}+3=0$$$$\Rightarrow (\mathrm{m}-1)(\mathrm{m}-3)=0\{\begin{array}{l}\mathrm{m}=1\Rightarrow \lambda =3\\ \mathrm{m}=3\Rightarrow \lambda =1\end{array}$$$$\lambda =\{1,3\}\divideontimes$$

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