Normal Vector to Ellipsoid at Point [1, 1, 1]

How can we compute the normal vector to the ellipsoid at the point [1, 1, 1]?

The normal vector to the ellipsoid at the point [1, 1, 1] is [2, 2, 4]. How is this normal vector calculated?

To compute the normal vector to the ellipsoid at the point [1, 1, 1], we need to take the gradient of the function that defines the ellipsoid. In this case, the function is x² + y² + 2z² = 4.

The normal vector to the ellipsoid at a given point can be computed by taking the gradient of the function that defines the ellipsoid. In this case, the function is x² + y² + 2z² = 4.

To find the normal vector at the point [1, 1, 1], we need to compute the gradient of the function at that point. The gradient is a vector that points in the direction of the steepest increase of the function at a given point.

By taking the partial derivatives of the function x² + y² + 2z² with respect to x, y, and z, we get [2x, 2y, 4z]. Substituting the coordinates of the point [1, 1, 1] into the gradient, we get [2, 2, 4]. Therefore, the normal vector to the ellipsoid at the point [1, 1, 1] is [2, 2, 4].

Understanding normal vectors and how to compute them is essential in various mathematical applications and fields such as physics, engineering, and computer graphics. They provide valuable information about the direction perpendicular to a surface at a given point, which is crucial in solving problems related to surfaces and shapes.