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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 $4x - 2y + 3z - 21 = 0.$
See the step by step solution

## Step 1: Find the normal vector

First, we identify the normal vector of the given plane, which is $$(4,-2,3)$$. Since our required plane is parallel to this given plane, it will have the same normal vector.

## Step 2: Use the point-normal form of the plane

Recall that the point-normal form of a plane is given by $$a(x-x_0) + b(y-y_0) + c(z-z_0) = 0$$, where $$(a,b,c)$$ is the normal vector of the plane and $$(x_0,y_0,z_0)$$ is a point on the plane. Now substitute the given point $$(4,-1,1)$$ and the normal vector $$(4,-2,3)$$ into the equation: $$4(x-4) - 2(y+1) + 3(z-1) = 0$$

## Step 3: Simplify the equation

Simplify the equation to get the final answer: \$4(x-4) - 2(y+1) + 3(z-1) = 0 \\ 4x - 16 - 2y - 2 + 3z - 3 = 0\) So, the equation of the plane is: $$4x - 2y + 3z - 21 = 0$$

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