find the unit normal vector to the surface x²+y²-2x+3=0 at the point (1, 2,-1) using app binary​

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).