A-plane-has-equation-x-z-4-3-The-line-l-has-vector-equation-r-cos-3-i-2-sin-j-cos-3-k-where-is-a-scalar-parameter-If-l-meets-the-plane-at-P-show-that-as-varies-P-describes

FacebookTweetPin

Question Number 1851 by 112358 last updated on 14/Oct/15

$$Aplanehasequationx-z=4\sqrt{3}.$$$$Thelinelhasvectorequation$$$$\mathit{r}=\lambda [(cos\theta +\sqrt{3})\mathit{i}+\left(\sqrt{2}sin\theta \right)\mathit{j}+(cos\theta -\sqrt{3})\mathit{k}]$$$$where\lambda isascalarparameter.$$$$IflmeetstheplaneatP,showthat,$$$$as\theta varies,Pdescribesacircle.$$

Answered by 123456 last updated on 15/Oct/15

$$\mathit{r}=\lambda [(\cos \theta +\sqrt{3})\mathit{i}+\sqrt{2}\sin \theta \mathit{j}+(\cos \theta -\sqrt{3})\mathit{k}]$$$$x-z=4\sqrt{3}$$$$\lambda (\cos \theta +\sqrt{3})-\lambda (\cos \theta -\sqrt{3})=4\sqrt{3}$$$$2\lambda \sqrt{3}=4\sqrt{3}$$$$\lambda =2$$$$\dots ..$$$$\mathrm{rotation}\mathrm{of}\mathrm{axis}$$$$\left[\begin{array}{c}x\\ z\end{array}\right]=\left[\begin{array}{cc}\cos 45°& -\sin 45°\\ \sin 45°& \cos 45°\end{array}\right]\left[\begin{array}{c}{x}_r\\ z_r\end{array}\right]$$$$\left[\begin{array}{cc}\cos 45°& \sin 45°\\ -\sin 45°& \cos 45°\end{array}\right]\left[\begin{array}{c}x\\ z\end{array}\right]=\left[\begin{array}{c}{x}_r\\ z_r\end{array}\right]$$$$\dots ..$$$$\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}\lambda (\cos \theta +\sqrt{3})\\ \lambda \sqrt{2}\sin \theta \\ \lambda (\cos \theta -\sqrt{3})\end{array}\right]$$$$\left[\begin{array}{c}{x}_r\\ y_r\\ z_r\end{array}\right]=\left[\begin{array}{c}\lambda \sqrt{2}\cos \theta \\ \lambda \sqrt{2}\sin \theta \\ -\lambda \sqrt{6}\end{array}\right]$$$$\dots \dots$$$$x_r^2+y_r^2=2{\lambda }^2\left(\mathrm{equation}\mathrm{of}\mathrm{circle}\right)$$

Terms of Service
Privacy Policy
Contact: info@tinkutara.com

FacebookTweetPin