Write an equation of the plane with normal vector n =⟨−7,−4,−6⟩ passing through the point (−8,4,6) in scalar form in the form of equation = 20
Question
Write an equation of the plane with normal vector
passing through the point in scalar form
in the form of equation
Solution
The equation of a plane in 3D space is given by the formula:
Ax + By + Cz = D
where A, B, and C are the components of the normal vector to the plane, and D is a constant. The point (x0, y0, z0) lies on the plane.
Given the normal vector n = ⟨−7,−4,−6⟩ and the point (−8,4,6), we can substitute these values into the equation:
-7(x - (-8)) - 4(y - 4) - 6(z - 6) = 0
Simplify this to get:
-7x + 56 - 4y + 16 - 6z + 36 = 0
Combine like terms to get:
-7x - 4y - 6z + 108 = 0
We want the equation in the form of equation = 20, so subtract 108 from both sides to get:
-7x - 4y - 6z = -88
This is the equation of the plane in scalar form.
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