Problem 33
Question
Find an equation of the plane parallel to the plane \(Q\) passing through the point \(P_{0}\). $$Q: 4 x+3 y-2 z=12 ; P_{0}(1,-1,3)$$
Step-by-Step Solution
Verified Answer
Answer: The equation of the parallel plane passing through point \(P_0(1, -1, 3)\) is: \(4x + 3y - 2z = -5\).
1Step 1: Identify the normal vector
Since plane Q is given by the equation \(4x + 3y - 2z = 12\), the coefficients of x, y, and z are 4, 3, and -2, respectively. Thus, the normal vector (NV) for plane Q is: \(NV = \langle 4, 3, -2 \rangle\).
2Step 2: Substitute point P0 into the equation for the parallel plane
Since the desired plane is parallel to plane Q, it also has the same normal vector. So, the equation of the parallel plane passing through the point \(P_0(1, -1, 3)\) will be in the form: \(4x + 3y - 2z = d\).
Now we need to find the value of d. Substitute the coordinates of point \(P_0\) into the equation: \(4(1) + 3(-1) - 2(3) = d \Rightarrow 4 - 3 - 6 = d \Rightarrow -5 = d\).
3Step 3: Write the equation of the parallel plane
Now that we have found the value of d, we can write the equation of the parallel plane using the normal vector and the constant term. The equation of the parallel plane passing through point \(P_0(1, -1, 3)\) is: $$4x + 3y - 2z = -5$$.
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