find the unit normal vector to the surface x²+y²-2x+3=0 at the point (1, 2,-1) using app binary
Question
find the unit normal vector to the surface x²+y²-2x+3=0 at the point (1, 2,-1) using app binary
Solution
The surface given is a cylinder and its equation can be rewritten as (x-1)² + y² = 0. This is a cylinder with axis parallel to the z-axis and passing through the point (1,0,0).
The normal vector to a surface F(x,y,z) = 0 at a point P is given by the gradient of F at P.
The gradient of F is a vector of the partial derivatives of F with respect to x, y, and z.
In this case, F(x,y,z) = x² + y² - 2x + 3, so the gradient of F is (2x-2, 2y, 0).
At the point (1,2,-1), the gradient of F is (21-2, 22, 0) = (0,4,0).
This is a normal vector to the surface at the point (1,2,-1).
However, a unit normal vector is a normal vector of length 1.
The length of the vector (0,4,0) is sqrt(0² + 4² + 0²) = 4.
So, the unit normal vector to the surface at the point (1,2,-1) is (0/4, 4/4, 0/4) = (0,1,0).
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