Find parametric equations for the line tangent to the curve of intersection of the surfaces at t

step 1 So we have two surfaces given to us. We have the first one that I'm going to refer to as f x squ

Find parametric equations for the line tangent to the curve...

Key Concepts

Parametric Equations for a Line

Parametric equations express the coordinates of points on a line as functions of a single parameter, typically written in the form x = x? + at, y = y? + bt, and z = z? + ct. This representation is essential in describing the direction and location of a line in space, making it easier to perform further analysis or computations, such as finding intersections or distances.

Tangent Vector

The tangent vector to a curve at a given point indicates the direction in which the curve is heading at that point. It provides the necessary directional information to define the tangent line and is computed either by differentiating a parametric representation of the curve or by using other techniques such as the cross product of gradients when the curve is defined by the intersection of surfaces.

Intersection of Surfaces

The curve that arises from the intersection of two surfaces consists of all points that satisfy the equations of both surfaces simultaneously. Studying the intersection involves understanding how the two surfaces interact in space, and various calculus techniques can be applied to determine properties of the resulting curve, such as its tangent direction at a given point.

Gradient and Cross Product

The gradient of a surface is a vector that points in the direction of the greatest rate of increase of a function and is perpendicular to the surface. When two surfaces intersect, taking the cross product of their gradient vectors at a point yields a vector that is tangent to the curve formed by their intersection. This method provides a systematic way to find the directional vector required for formulating the parametric equations of the tangent line.

Transcript

00:13 So we have two surfaces given to us.

00:17 We have the first one that i'm going to refer to as f x squared plus 2y plus 2z equals 2.

00:31 And the second surface is going to be referred to as g, and that's the surface, y equals 1.

00:38 Now these two surfaces intersect at a curve, and we want to find the tangent line to that curve at the point one, one, one half.

00:53 So first i have to find the gradient of each function by doing the partial derivative with respect to each of the variables and taking each of those partial derivatives, evaluating them at our point one, one, one half, and then taking that times our vector components in the direction of that variable.

01:14 So the gradient of f will be the partial derivative with respect to x, so y and 0 treated as constants, that gives us 2x times our vector i, which is the vector component in the x direction, plus 2 times vector j, partial derivative with respect to y, times the vector component in the y direction, plus 2 times vector k, because of the partial derivative with respect to y, times the vector component in the y direction, plus 2 times vector k, because of the partial derivative with respect to.

01:50 To z of our function.

01:53 And then we have to evaluate that at our point one one -half.

01:59 So we only have the x to plug the coordinate into.

02:04 So two times one is going to give us 2i plus 2j plus 2k.

02:17 For g the gradient of that we have to do the partial derivative of each variable.

02:23 Since there's no x and no z, those partial derivatives will equal zero, so there won't be an x and a z part of this vector.

02:31 And the partial derivative of y is just one times vector j.

02:37 So there's our gradient of g.

02:39 Now the direction our lines going, that vector, i'm going to call vector v, is the cross product of the gradient of f and the gradient of g.

02:52 And we find the cross -product.

02:54 And we find the by finding the determinant of 3x3 by our vector components i, j and k in the first row, and then the components of the gradient of f in the second row and the gradient of g in the third row and it's always done in that order...

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