 Suppose we have a surface z equals f of x, y. At any point x0, y0, z0, we can talk about the tangent line. Actually, we can't. There's an infinite number of tangent lines. But let's think about this. If a one-dimensional curve has a one-dimensional tangent line, then we should expect that a two-dimensional surface should have a two-dimensional tangent plane. And so the question you've got to ask yourself is, how do we find it? Let's think about this a little bit more. Suppose we take the surface z equals f of x, y, and find the tangent plane at a point. Now, if we intersect this plane with the plane, say, x equals x0, well, that's the intersection of two planes, and that will give us a straight line. Meanwhile, if we intersect the surface with the plane x equals x0, we get some sort of curve in the plane, and that curve will have a tangent line. So what if we require they be the same line? Well, let's try it out. Find the equation of the line tangent to the curve formed by the intersection of z equals x cubed minus y cubed, and the plane x equals 2 at y equals 3. So to clarify the geometry here, we have some sort of surface, and we're going to cut this surface with a plane, and that'll give us a one-dimensional curve. And I'm interested in finding the tangent line to that curve at a particular value of y. So to begin with, let's figure out where our point actually is. We're on the plane x equals 2. We want to know when y equals 3. So the only thing we don't know is the value of z. If only there was some formula that we had that would give us the value of z. Oh, wait. OK, here we have it. And so we find that z is equal to negative 19. And so we're at the point 2 because we're on the plane x equals 2, 3 because we want y equal to 3, and z equal negative 19 because that's where our z value is for x equals 2, y equals 3. And so our tangent line goes through the point 2, 3, negative 19. And so our equation of the line, well, first, we'll navigate to the point 2, 3, negative 19. So we're on the plane x equals 2. And so that means x will always equal 2. And one of the things you might remember from single variable calculus is if a value doesn't change, then we can use it before we do any calculus. So since x is always equal to 2, our equation then becomes. And this is an equation in two variables. So there's a curve. And I could talk about the line tangent to this curve. And the derivative will be, now, y is actually a variable. y could take on many different values, but we only care about what happens when y equals 3. And so we find our derivative. Now, remember, we're actually working in three dimensions. So let's think about this. The derivative dz dy equals negative 27, or negative 27 divided by 1. So remember that the derivative tells us how the variable z is changing as y changes. And negative 27 divided by 1 means that to move along the tangent line, we increase our y values by 1 unit while we decrease our z values by 27 units. Again, we're in three-dimensional space, so we know what happens to y and to z. What about x? And again, it's useful to remember, since we're in the plane x equals 2, x is always equal to 2, and so x doesn't change at all. And so to go from a point on the tangent line to another point on the tangent line, we'll follow along the vector, don't change x, increase y by 1, decrease z by 27 units. And so our tangent line goes to the direction 0, 1, negative 27, and we can go any scalar multiple of that direction vector. What if we considered intersecting the surface with the plane y equals 3? Well, in that case, we get a different tangent line. Let's take a look at that. So again, same curve, and we want x equals 2, y equals 3, z equals negative 19 still, but this time, instead of taking the plane x equals 2, we'll take the plane y equals 3, and that allows x and z to change, and so we'll want to know what happens at x equal to 2. So again, as long as we're staying on the plane y equals 3, y is always equal to 3. It's a constant, and we can replace it in our equation. We can differentiate. At x equals 2, we find dz dx is equal to, and remember the derivative is the rate of change, in this case of z with respect to x. So to stay on the line with dz dx equal 12 or 12 divided by 1 in the plane y equals 3, we need to increase x by 1 while we increase z by 12 and leave y unchanged, and so that means we're going to move along the vector 1, 0, 12. So remember, our tangent line will go through the 0.2, 3, negative 19, and it will go in the direction of the vector 1, 0, 12, so its equation will be. So let's think about what we've done so far. We have a line formed by the intersection of the tangent plane and a plane parallel to the yz plane. We also found a line formed by the intersection of the tangent plane and a plane parallel to the xz plane, and we want to find the equation of the plane that includes both lines. But we know the vector is giving the direction of both lines and the vector perpendicular to both will be orthogonal to the plane and we can find this vector using the cross product. So let's actually find that plane. So we know the equations of both lines. So since the vectors 0, 1, negative 27 and 1, 0, 12 are in the plane, a vector orthogonal to the plane is going to be the cross product. So we find the cross product. So if x, y, z is a point on the plane, then we'll want the vector from 2, 3, negative 19, that's one point in the plane, to the other will be orthogonal to our vector. And so we have the dot product equal to 0 and that gives us the equation of the tangent plane.