Problem 12 Find the equation of the plane w... [FREE SOLUTION]

Chapter 12: Problem 12

Find the equation of the plane which passes through the point \((2,-3,4)\) and is parallel to the plane \(2 x-5 y-7 z+15=0\).

Short Answer

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Question: Find the equation of the plane that passes through the point (2, -3, 4) and is parallel to the plane 2x - 5y - 7z + 15 = 0. Answer: The equation of the plane that passes through the point (2, -3, 4) and is parallel to the given plane is 2x - 5y - 7z + 18 = 0.

Step by step solution

Extract information from the problem

We are given a point \(P(2,-3,4)\) through which the plane passes, and that the plane is parallel to the given plane \(2x - 5y - 7z + 15 = 0\). From the given plane equation, we can identify the normal vector \(\bold{n}\) as \(\langle 2, -5, -7 \rangle\).

Use the point-normal form of a plane's equation

The point-normal form of a plane's equation is given by: \(\bold{n} \cdot (\bold{x} - \bold{p}) = 0\), where \(\bold{n}\) is the normal vector, \(\bold{x}\) is the position vector of any point on the plane, and \(\bold{p}\) is the position vector of the given point. In our case, we have \(\bold{n} = \langle 2, -5, -7 \rangle\) and \(\bold{p} = \langle 2, -3, 4 \rangle\). Let the position vector of any point on the plane be \(\bold{x} = \langle x, y, z \rangle\). Then: \(\bold{x} - \bold{p} = \langle x - 2, y + 3, z - 4 \rangle\) Now we have to find the dot product of \(\bold{n}\) and \(\bold{x} - \bold{p}\) to get the equation of the plane.

Calculate the dot product and equation of the plane

The equation of the plane can be found by calculating the dot product of \(\bold{n}\) and \(\bold{x} - \bold{p}\): \((\langle2,-5,-7\rangle)\cdot(\langle x-2,y+3,z-4\rangle) = 0\) Calculating the dot product, we get: \(2(x-2) - 5(y+3) - 7(z-4) = 0\) Finally, expanding and simplifying the equation: \(2x - 5y - 7z + 18 = 0\) So, the equation of the plane that passes through the point \((2,-3,4)\) and is parallel to the given plane is: \(\boxed{2x - 5y - 7z + 18 = 0}\).

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91影视

Find the equation of the plane which passes through the point \((2,-3,4)\) and is parallel to the plane \(2 x-5 y-7 z+15=0\).

Short Answer

Step by step solution

Extract information from the problem

Use the point-normal form of a plane's equation

Calculate the dot product and equation of the plane

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