This page covers one application of gradient vectors and directional derivatives, to find normal and tangent lines and planes.
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To determine the equation of a plane, we need one point and a normal vector. Since the gradient vector is normal to the level curves (and also to the tangent plane), we have a normal vector and we are usually given a point in the problem statement. This is enough information to determine the equation of the tangent plane.
Here is a video clip with a great explanation of this idea. He also explains how using the gradient notation simplifies the notation significantly.
video by Dr Chris Tisdell 

Practice
Find the tangent plane to the surface \( P(x,y,z) = x^2+y^2z^39 \) at the point \( (3,0,0) \)
Problem Statement
Find the tangent plane to the surface \( P(x,y,z) = x^2+y^2z^39 \) at the point \( (3,0,0) \)
Solution
video by Thomas Wernau 

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Find the equation of the tangent plane and the symmetric equations of the normal line to the surface \( 2(x2)^2 + (y1)^2 + (x3)^2 = 10 \) at the point \( (3,3,5) \).
Problem Statement
Find the equation of the tangent plane and the symmetric equations of the normal line to the surface \( 2(x2)^2 + (y1)^2 + (x3)^2 = 10 \) at the point \( (3,3,5) \).
Solution
video by Krista King Math 

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Determine a normal vector and the equation of the tangent plane to the surface \( z = x^2 + 2y^2 \) at the point \( A(2,1,6) \).
Problem Statement
Determine a normal vector and the equation of the tangent plane to the surface \( z = x^2 + 2y^2 \) at the point \( A(2,1,6) \).
Solution
video by Dr Chris Tisdell 

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Find the tangent plane to the surface \( x=u^2, y=uv^2, z=v^2 \) for \( u,v \geq 0 \) at the point \((1,0,1)\).
Problem Statement
Find the tangent plane to the surface \( x=u^2, y=uv^2, z=v^2 \) for \( u,v \geq 0 \) at the point \((1,0,1)\).
Solution
video by Michael Hutchings 

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Calculate the tangent plane and the normal line to the surface \( x^2 + y^2 + z = 9 \) at the point \( (1,2,4) \).
Problem Statement
Calculate the tangent plane and the normal line to the surface \( x^2 + y^2 + z = 9 \) at the point \( (1,2,4) \).
Solution
video by Dr Chris Tisdell 

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Calculate the gradient, directional derivative and equation of the tangent plane of \( g(x,y,z) = \sin(xyz) \) at the point \( (\pi, 1/2, 1/2) \) in the direction \( \vec{v} = \langle \sqrt{3},2,3 \rangle \).
Problem Statement
Calculate the gradient, directional derivative and equation of the tangent plane of \( g(x,y,z) = \sin(xyz) \) at the point \( (\pi, 1/2, 1/2) \) in the direction \( \vec{v} = \langle \sqrt{3},2,3 \rangle \).
Solution
video by Dr Chris Tisdell 

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Calculate the equation of the tangent plane to \( x^4 + xy + y^2 = 19 \) at the point \( (2,3) \).
Problem Statement
Calculate the equation of the tangent plane to \( x^4 + xy + y^2 = 19 \) at the point \( (2,3) \).
Solution
video by Krista King Math 

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Calculate the equation of the tangent plane to the surface \(f(x,y)=2y\cos(5x3y)\) at the point \((3,5,10)\).
Problem Statement
Calculate the equation of the tangent plane to the surface \(f(x,y)=2y\cos(5x3y)\) at the point \((3,5,10)\).
Solution
video by MIP4U 

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Find the normal line to the surface \(f(x,y)=x^25xy+2\) at the point \((1,1,1)\)
Problem Statement
Find the normal line to the surface \(f(x,y)=x^25xy+2\) at the point \((1,1,1)\)
Solution
video by Thomas Wernau 

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Find a set of symmetric equations for the normal line to the surface \(xyz=12\) at \((2,2,3)\)
Problem Statement
Find a set of symmetric equations for the normal line to the surface \(xyz=12\) at \((2,2,3)\)
Solution
video by Thomas Wernau 

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Find a set of symmetric equations for the normal line to the surface \(F(x,y)=x^2y^2+1\) at the point \((1,1,1)\).
Problem Statement
Find a set of symmetric equations for the normal line to the surface \(F(x,y)=x^2y^2+1\) at the point \((1,1,1)\).
Solution
video by Thomas Wernau 

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Find the set of symmetric equations for the normal line to the surface \(x^2+2y^2 = 20\) at \((0,1,3)\).
Problem Statement
Find the set of symmetric equations for the normal line to the surface \(x^2+2y^2 = 20\) at \((0,1,3)\).
Solution
video by Thomas Wernau 

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