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Problem 893

Find the equation of the plane passing through the point \((4,-1,1)\) and parallel to the plane \(4 x-2 y+3 z-5=0\)

Expert verified

The equation of the plane passing through the point \((4,-1,1)\) and parallel to the given plane is \[4x - 2y + 3z - 21 = 0.\]

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Chapter 51

Show that the point \(\mathrm{P}_{1}(2,2,3)\) is equidistant from the points \(\mathrm{P}_{2}(1,4,-2)\) and \(\mathrm{P}_{3}(3,7,5)\)

Chapter 51

Find an equation of the sphere which has the segment joining \(\mathrm{P}_{1}(2,-2,4)\) and \(\mathrm{P}_{2}(4,8,-6)\) for a diameter.

Chapter 51

Show that the points \(\mathrm{A}(-1,-3,7), \mathrm{B}(-2,-2,9)\), and \(\mathrm{C}(1,3,5)\) are the vertices of a right triangle.

Chapter 51

Describe the geometric shape determined by the graph of the equation \(\mathrm{x}^{2}+\mathrm{y}^{2}=1\) when plotted on a 3 -dimensional system of rectangular coordinates.

Chapter 51

Given points $P_{1}\left(x_{1}, y_{1}, z_{1}\right), P_{2}\left(x_{2}, y_{2}, z_{2}\right)\( and the origin \)0(0,0,0)$ (a) show that $\cos \angle \mathrm{P}_{1} \mathrm{OP}_{2}=\left(\mathrm{x}_{1} \mathrm{x}_{2}+\mathrm{y}_{1} \mathrm{y}_{2}+\mathrm{z}_{1} \mathrm{z}_{2}\right) /\left(\mathrm{d}_{1} \mathrm{~d}_{2}\right)$ where \(\mathrm{d}_{1}=\mathrm{OP}_{1}\) and \(\mathrm{d}_{2}=0 \mathrm{P}_{2}\) (b) Find a condition on $\mathrm{x}_{1}, \mathrm{x}_{2}, \mathrm{y}_{1}, \mathrm{y}_{2}, \mathrm{z}_{1}\(, and \)\mathrm{z}_{2}$ such that $\underline{P}_{\underline{1}} \underline{\underline{O}} \perp \underline{\mathrm{P}}_{\underline{2}} \underline{\mathrm{O}} .$

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