Find a point on the surface z=8-3 x^2-2 y^2 at which the tangent plane is perpendicular to the l

step 1 In this problem, we're trying to find a point on the surface, the given surface here, that is pe

Find a point on the surface z=8-3 x^2-2 y^2 at which the...

Key Concepts

Gradient

The gradient of a scalar function is a vector that points in the direction of greatest increase of the function and its magnitude indicates the rate of change. In the context of surfaces defined implicitly or as graphs, the gradient at a point is perpendicular to the level surface, making it a natural candidate for the normal vector to the surface at that point.

Tangent Plane

A tangent plane to a surface at a given point is the plane that best approximates the surface in the neighborhood of that point. Its normal vector is often found by taking the gradient of the function that defines the surface. The tangent plane is critical for understanding local linear behavior of the surface.

Normal Vector and its Relation to Perpendicularity

The normal vector to a surface is a vector that is perpendicular to the tangent plane at a given point. When requiring a tangent plane to be perpendicular to a line, one uses the fact that if a line is perpendicular to the plane, then the line’s direction vector must be parallel to the normal vector. Establishing this relationship typically involves setting the normal vector equal to a scalar multiple of the line’s direction vector.

Line Direction Vector

A line’s direction vector specifies the orientation of the line in space. It is derived from the line’s parametric equations and is essential when comparing the line’s orientation to other geometric objects, such as when determining the perpendicularity or parallelism between the line and a plane.

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