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Question Number 133199 by rexford last updated on 19/Feb/21

Answered by mr W last updated on 20/Feb/21

$$\overrightarrow{\mathit{A}\mathit{B}}=(-1,5,-3)$$$$\overrightarrow{\mathit{A}\mathit{C}}=(-4,3,3)$$$$\overrightarrow{\mathit{A}\mathit{D}}=(1,7,\lambda +1)$$$$1\times (15+9)-7\times (-3-12)+(\lambda +1)(-3+20)=0$$$$24+7\times 15+17(\lambda +1)=0$$$$\Rightarrow \lambda =-\frac{146}{17}$$

Commented by liberty last updated on 20/Feb/21

$$\mathrm{typo}.24+105+17=105+41=146$$

Commented by mr W last updated on 20/Feb/21

$$yes,thanks!$$

Commented by rexford last updated on 20/Feb/21

$$Thanks$$

Answered by john_santu last updated on 20/Feb/21

$$\mathrm{let}\overrightarrow{\mathrm{n}}=\overrightarrow{a}\times \overrightarrow{b}\mathrm{be}\mathrm{a}\mathrm{normal}\mathrm{vector}\mathrm{plane}$$$$\mathrm{then}\overrightarrow{\mathrm{n}}=\left|\begin{array}{c}3-2-1\\ 23-4\end{array}\right|=11\hat{\mathrm{i}}+10\hat{\mathrm{j}}+13\hat{\mathrm{k}}$$$$\mathrm{since}\mathrm{the}\mathrm{points}\mathrm{A},\mathrm{B},\mathrm{C}\mathrm{and}\mathrm{D}$$$$\mathrm{so}\overrightarrow{\mathrm{n}}.\mathbf{CD}=0;\mathrm{where}\mathbf{CD}=(5,4,\lambda -2)$$$$\iff 55+40+13\lambda -26=0$$$$\iff \lambda =\frac{26}{99}$$

Answered by EDWIN88 last updated on 20/Feb/21

Answered by liberty last updated on 20/Feb/21

$$\mathrm{normal}\mathrm{vector}\mathrm{plane}$$$$\overrightarrow{\mathrm{n}}=\mathrm{AB}\times \mathrm{AC}=\left|\begin{array}{c}-15-3\\ -433\end{array}\right|$$$$\overrightarrow{\mathrm{n}}=24\hat{\mathrm{i}}+15\hat{\mathrm{j}}+17\hat{\mathrm{k}}$$$$\mathrm{put}\mathrm{AD}=\hat{\mathrm{i}}+7\hat{\mathrm{j}}+(\lambda +1)\hat{\mathrm{k}}$$$$\mathrm{so}\mathrm{we}\mathrm{get}\mathrm{equation}\mathrm{of}\mathrm{plane}\mathrm{is}$$$$\overrightarrow{\mathrm{n}}.\mathrm{AD}=0$$$$\Rightarrow 24+105+17\lambda +17=0$$$$\Rightarrow \lambda =-\frac{146}{17}$$

Commented by rexford last updated on 20/Feb/21

$$Thankyouverymuch$$

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