Find a scalar equation of the plane that contains the given point P=(-3,-3,1)and the given normal vector n = (-1,4,7)
To find a scalar equation of a plane, we can use the point-normal form equation of a plane. The point-normal form equation of a plane is given by:
A(x - x₁) + B(y - y₁) + C(z - z₁) = 0
Where (x₁, y₁, z₁) is a point on the plane and (A, B, C) is the normal vector of the plane.
In this case, we have the point P=(-3,-3,1) and the normal vector n = (-1,4,7). Substituting these values into the point-normal form equation of the plane, we can write:
(-1)(x - (-3)) + (4)(y - (-3)) + (7)(z - 1) = 0
Simplifying this equation, we have:
-(x + 3) + 4(y + 3) + 7(z - 1) = 0
Now, let's multiply out the terms:
-x - 3 + 4y + 12 + 7z - 7 = 0
Combining like terms, we get:
-x + 7z + 4y + 2 = 0
This is the scalar equation of the plane that contains the point P=(-3,-3,1) and the normal vector n=(-1,4,7).