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### Course: Multivariable calculus>Unit 3

Lesson 1: Tangent planes and local linearization

# Controlling a plane in space

How can you describe a specified plane in space as the graph of a function? Created by Grant Sanderson.

## Want to join the conversation?

• This is essentially the point slope form but for a plane in 3D space instead of a line, and the values in the parentheses represent how the plane is going to shift in space
• wrong question we on planing it not 3d space
(1 vote)
• This is asthoninglishy hard to wrap my head around.
• How does this equation of a plane correspond with the

n1*x + n2*y + n3*z = n.p

Equation, where n is a normal vector to the plane with components <n1, n2, n3>, p is some fixed point on the plane (or the position vector pointing to the point), and the n.p is the dot product of n and p?
(1 vote)
• It is the same equation but just rewritten,

notice that you have
n1*x + n2*y + n3*z = n.p
n.r = n.p
n.r - n.p = 0
n.(r-p) =
(n1,n2,n3) . (x-x0, y-y0, z-z0) = 0
n1*(x-x0) + n2*(y-y0) + n3*(z-z0) = 0

What remains to see is that n = (dL/dx , dL/dy, -1) and that z = L(x,y).

1.) To see this I copied Evgenii Neumerzhitckii answer from another video:

let a = dL/dx and b = dL/dy and
Write the Equation in the video as:
z = a(x - x0) + b(y - y0) + z0,

Rearrange, to get the plane equation in standard form:
ax + by - z = -z0 + a*x0 + b*y0.

As we know from linear algebra, the coefficients of x, y, z are the coordinates of the normal vector:
n = (a, b, -1).

2.) Another method which might be confusing is that you can find two vectors that lie on the tangent plane using the partial derivatives of the original function, they are (0,1, dF/dy) and (1,0,dF/dx) at a specific point, this is true because a partial derivative with respect to x gives us the tangent slope for the curve resulting from intersecting an xz plane with the graph.

The cross product between these two vectors gives us a normal vector which is (dF/dx, dF/dy, -1), we know that dF/dx = dL/dx at a specific point, this is the case because it is a tangent plane at that point. so n = (dL/dx , dL/dy, -1).

• At , why is it that the any plane intersecting the other plane results in the exact same slope?
• It isn't any plane, but parallel planes (y=1, y=3, y=5, etc) intersecting the given plane.
Two non-parallel and infinitely extending planes always intersect in a straight line, and the angle between the intersecting planes is given by the angle between the normal vectors to the planes. Since a normal vector to a plane is also normal (i.e., orthogonal) to a parallel plane, the angle of intersection doesn't change when a given plane is intersected by a plane parallel to another intersecting plane. So all these intersections by parallel planes on a given plane produce parallel straight lines; that is, these intersections produce straight lines with the same slope.
• Why L(x,y) and not L (x,y,z)? The video is 3-d..
(1 vote)
• The graph is 3d, but the function is operating on a 3d input space! L(x,y) is the correct writing because it's the function defining the plane to which the function looks very similar on an infinitesimal scale around the point (x,y).
• God, what does he talking about?
(1 vote)
• Hi,
I don't understand the intuition behind the shifting in the equation.
Why do we substract/add a constant to the variable (a(x-xo) + b(y-yo) + c instead of ax + by + c) ?
(1 vote)
• how do you know when you fou tHow does this equation of a plane correspond with the

n1*x + n2*y + n3*z = n.p

Equation, where n is a normal vector to the plane with components <n1, n2, n3>, p is some fixed point on the plane (or the position vector pointing to the point), and the n.p is the dot product of n and p?e parchle divide
(1 vote)
• How did he expect ‘a’ would be 2 and ‘y’ would be 1? He just see the whole plane and guess roughly?
(1 vote)
• What does he mean when he says the function should have a plane as its graph? What is graph of planes?