Using the previous theorems, we shall specify a method in finding the tangent plane to a level surface.
Remember that although we can’t sketch the function
, we can sketch a level surface of that function given by 
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CONCEPT
We will let
 as our function to study. The level surface we will use is  which gives us the cone
We then use the del operator on
where Other than the origin, the gradient
Explicitly, we can also show that these two vectors are indeed perpendicular by calculating their dot product.
since on the surface, We can use the gradient vector to find the tangent plane to the level surface We know that
After multiplying, this gives us the equation
which turns out to be our equation for the tangent plane at All information presentated, less questions and exercises, is original content of Donny, with slight references to various books.
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can be drawn as an arrow point into the cone and perpendicular to the side of the cone at any one point. 
is perpendicular to the position vector 
at 
as shown below.
.
at any point 
where the gradient is not 0 and is defined. This can be done by simple vector algebra.
is on the tangent plane and vector 
is a vector parallel to the tangent plane and hence is perpendicular to 
and so
.
