Let P(x, y, z) be a point in the first octant, whose projection in the x y-plane is the po…

Question

Let $P(x, y, z)$ be a point in the first octant, whose projection in the $x y$-plane is the point $Q$. Let $$O P=\gamma$$; the angle between $O Q$ and the positive $x$-axis be $$\theta$$; and the angle between $O P$ and the posi...

Accepted Answer

$\begin{gathered} \overset{	o \leftarrow }{OP}=x\hat{i}+y\hat{j}+z\hat{k} \ \overset{	o \leftarrow }{OQ}=x\hat{i}+y\hat{j} \ |OP|=\gamma =\sqrt{{x}^{2}+{y}^{2}+{z}^{2}} \ \cos 	heta =\frac{x}{\sqrt{{x}^{2}+{y}^{2}}}\Rightarrow {\cos }^{2}	heta =\frac{{x}^{2}}{{\gamma }^{2}-{z}^{2}}=\frac{{x}^{2}}{{\gamma }^{2}-{\gamma }^{2}{\cos }^{2}\phi } \ \cos \phi =\frac{z}{\sqrt{{x}^{2}+{y}^{2}+{z}^{2}}}=\frac{z}{\gamma } \ Px=\sqrt{{y}^{2}+{z}^{2}} \ d=\sqrt{{\gamma }^{2}-{x}^{2}} \ \Rightarrow {x}^{2}={\gamma }^{2}{\sin }^{2}\phi {\cos }^{2}	heta \ \Rightarrow d=\sqrt{{\gamma }^{2}-{\gamma }^{2}{\sin }^{2}\phi {\cos }^{2}	heta } \ =\gamma \sqrt{1-{\sin }^{2}\phi {\cos }^{2}	heta } \end{gathered}$

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