Final answer:
To find the equation of the plane, construct vectors from the given points, calculate their cross product for a normal vector, and use this vector with a point to derive the equation. The equation of the plane is -13x + 17y + 7z + 42 = 0.
Step-by-step explanation:
To find the equation of the plane through the points (3, -1, 2), (8, 2, 4), and (-1, -2, -3), we will use the following steps.
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Vector A (using the first two points) = (8 - 3, 2 - (-1), 4 - 2) = (5, 3, 2).
Vector B (using the first and third points) = (-1 - 3, -2 - (-1), -3 - 2) = (-4, -1, -5).
The cross product of vectors A and B gives us vector N:
N = A x B = |i j k|
|5 3 2|
|-4 -1 -5|
= (3*(-5) - 2*(-1))i - (5*(-5) - 2*(-4))j + (5*(-1) - 3*(-4))k
= (-15 + 2)i - (-25 + 8)j + (-5 + 12)k
= -13i + 17j + 7k
Therefore, vector N = (-13, 17, 7).
Now, we use vector N to write the plane equation with point (3, -1, 2):
-13(x - 3) + 17(y + 1) + 7(z - 2) = 0
-13x + 39 + 17y + 17 + 7z - 14 = 0
-13x + 17y + 7z + 42 = 0
Thus, the equation of the plane is -13x + 17y + 7z + 42 = 0.