Find a vector equation for the tangent line to the curve of...

Key Concepts

Vector Equation of a Line

A vector equation of a line expresses every point along the line as the sum of a fixed point vector and a scalar multiple of a direction vector. This formulation is useful for describing lines in space in a compact and efficient way, allowing one to easily compute points along the line.

Tangent Vector

A tangent vector to a curve at a point provides the direction in which the curve is proceeding at that specific point. It is used to construct the tangent line and reflects the instantaneous direction of motion along the curve.

Curve of Intersection of Surfaces

The curve of intersection of two surfaces is the set of points that simultaneously satisfy the equations defining both surfaces. This concept is important as it involves finding a common geometric feature arising from multiple constraints.

Gradient Vector

The gradient vector of a function representing a surface points in the direction of maximum increase of the function and is orthogonal to the level surface at that point. It is essential for determining directions perpendicular to a surface, which in turn helps in finding vectors tangent to the curve of intersection.

Cross Product Method

Using the cross product of two non-parallel gradient vectors of intersecting surfaces yields a vector that is perpendicular to both gradients. This resultant vector lies in the tangent planes of the surfaces and thus serves as a tangent vector to the curve of intersection.

Transcript

00:02 We are given two cylinders and a point which lies on the intersection of the two cylinders.

00:13 And we are asked to find a vector equation for the tangent line to the curve of intersection of these cylinders at this point.

00:25 The two cylinders have equations x squared plus y squared equals 25 and y squared plus c squared equals 20.

00:33 And the point lying on the curve of intersection is three, four, two.

00:45 In order to find the curve of intersection of these cylinders, we use substitution and elimination.

01:11 So we have that from our first equation, y squared is equal to 25 minus x squared.

01:29 And our second equation tells us that z squared is equal to 20 minus y squared, which is equal to to 20 minus 25 plus x squared or x squared minus 5.

01:59 So now we have y squared and z squared in terms of x.

02:05 Notice that at our point 342, both y and z are greater than 0.

02:13 So we have that the part of the curve we're interested in is the one for which y is equal to the positive square root of 25 minus 2.

02:24 X squared z is equal to the positive square root of x squared minus five and so we take x to be equal to t and the parameterization for this curve of intersection could be r of t equals t squared square root of 25 minus t squared and square root of t squared of t squared and square root of t squared minus 5, where we have that t is going to lie between, we have to have x squared minus 5 is greater than equal to 0, which means that has to lie between negative root 5 and positive root 5.

04:06 This was a mistake actually.

04:08 Instead of t being between negative root 5 and positive root 5, we have that t must be greater than or equal to root 5, or t must be less than or equal to negative root 5.

04:23 Root 5...

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