If from a point P(a,b,c) perpendicular PA and PB are drawn to YZ and ZX planes. Find the vector equation of the plane OAB.

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Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{We} \mathrm{have}: \\ \mathrm{P}\left(\mathrm{a},\mathrm{b},\mathrm{c}\right) \\ \mathrm{On} \mathrm{YZ}\hspace{0.17em}\mathrm{plane} \mathrm{X}\hspace{0.17em}\mathrm{is} 0. \\ \mathrm{Hence} \mathrm{coordinates} \mathrm{of} \mathrm{A}\hspace{0.17em}\mathrm{are} \left(0,\mathrm{b},\mathrm{c}\right) \\ \mathrm{On} \mathrm{ZX} \mathrm{plane} \mathrm{Y} \mathrm{is} 0. \\ \mathrm{Hence} \mathrm{coordinates} \mathrm{of} \mathrm{B} \mathrm{are} \left(\mathrm{a},0,\mathrm{c}\right) \\ \mathrm{Coordinates} \mathrm{of} \mathrm{Origin} \mathrm{i}.\mathrm{e} \mathrm{O} \mathrm{are} \left(0,0\right) \\ \Rightarrow \overrightarrow{\mathrm{OA}}=\left(0-0\right)\hat{\mathrm{i}}+\left(\mathrm{b}-0\right)\hat{\mathrm{j}}+\left(\mathrm{c}-0\right)\hat{\mathrm{k}}=\mathrm{b}\hat{\mathrm{j}}+\mathrm{c}\hat{\mathrm{k}} \\ \overrightarrow{\mathrm{OB}}=\left(\mathrm{a}-0\right)\hat{\mathrm{i}}+\left(0-0\right)\hat{\mathrm{j}}+\left(\mathrm{c}-0\right)\hat{\mathrm{k}}=\mathrm{a}\hat{\mathrm{i}}+\mathrm{c}\hat{\mathrm{k}} \\ \mathrm{Direction} \mathrm{normals} \mathrm{of} \mathrm{plane} \mathrm{will} \mathrm{be} \mathrm{given} \mathrm{by} \mathrm{cross} \mathrm{product} \mathrm{of} \overrightarrow{\mathrm{OA}} \mathrm{and} \overrightarrow{\mathrm{OB}}. \\ \overrightarrow{\mathrm{OA}}\times \overrightarrow{\mathrm{OB}}=\left|\begin{array}{ccc}\hat{\mathrm{i}}& \hat{\mathrm{j}}& \hat{\mathrm{k}}\\ 0& \mathrm{b}& \mathrm{c}\\ \mathrm{a}& 0& \mathrm{c}\end{array}\right| \\ =\left(\mathrm{bc}-0\right)\hat{\mathrm{i}}-\left(0-\mathrm{ac}\right)\hat{\mathrm{j}}+\left(0-\mathrm{ab}\right)\hat{\mathrm{k}} \\ =\left(\mathrm{bc}\right)\hat{\mathrm{i}}+\left(\mathrm{ac}\right)\hat{\mathrm{j}}-\left(\mathrm{ab}\right)\hat{\mathrm{k}} \\ \mathrm{As} \mathrm{plane} \mathrm{passes} \mathrm{through} \mathrm{origin}, \\ \mathrm{Hence} \mathrm{vector} \mathrm{equation} \mathrm{of} \mathrm{plane} \mathrm{will} \mathrm{be}: \\ \overrightarrow{\mathrm{r}}.\left[\left(\mathrm{bc}\right)\hat{\mathrm{i}}+\left(\mathrm{ac}\right)\hat{\mathrm{j}}-\left(\mathrm{ab}\right)\hat{\mathrm{k}}\right]+0=0 \\ \Rightarrow \overrightarrow{\mathbf{r}}\mathbf{.}\left[\left(\mathbf{bc}\right)\hat{\mathbf{i}}\mathbf{+}\left(\mathbf{ac}\right)\hat{\mathbf{j}}\mathbf{-}\left(\mathbf{ab}\right)\hat{\mathbf{k}}\right]\mathbf{=}\mathbf{0} \end{gathered}$$

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