# Tangents to parametric equations

In summary, the website discusses the concept of the unit tangent vector for a curve in space. This vector is obtained by finding the tangent vector of the curve and normalizing it. The tutorial also explains how to find the tangent line at a given point on the curve and clarifies the difference between tangent lines and tangent planes. While there are infinite tangent lines at a point on a curve, there is only one line in the tangent plane that has the same direction as the curve and this is the unit tangent vector.
Gold Member
The site http://tutorial.math.lamar.edu/Classes/CalcII/TangentNormalVectors.aspx talks of "the" unit tangent vector of r(t) = f(t)*i(t)+g(t)*j(t)+h(t)*k(t) as finding "the" tangent vector r'(t) = f'(t)*i(t)+g'(t)*j(t)+h'(t)*k(t) and normalizing it, and further with finding "the" tangent line at t=t0 as r(t0) + r'(t0)*t . (If I got that right.) But if one thinks of a tangent line as a line being perpendicular to the curve at the given point, then there are an infinite number of tangent lines (and unit tangent vectors), which is, as I understand it, the reason one deals with tangent planes in respect to curves in 3-D instead of tangent lines. What am I missing here?

Dr. Courtney
The site http://tutorial.math.lamar.edu/Classes/CalcII/TangentNormalVectors.aspx talks of "the" unit tangent vector of r(t) = f(t)*i(t)+g(t)*j(t)+h(t)*k(t) as finding "the" tangent vector r'(t) = f'(t)*i(t)+g'(t)*j(t)+h'(t)*k(t) and normalizing it, and further with finding "the" tangent line at t=t0 as r(t0) + r'(t0)*t . (If I got that right.) But if one thinks of a tangent line as a line being perpendicular to the curve at the given point, then there are an infinite number of tangent lines (and unit tangent vectors), which is, as I understand it, the reason one deals with tangent planes in respect to curves in 3-D instead of tangent lines. What am I missing here?
For your curve in space r(t) = f(t)i + g(t)j + h(t)k, at a given point on the curve P(x0, y0, z0), the tangent plane contains an infinite number of lines that lie in the plane and intersect the given point. However, there is only one line in this plane that has the same direction as the curve, and that's the one they're talking about in the tutorial. The unit tangent is a unit vector in that direction.

Ah, that makes sense. Thanks very much, Mark44

## 1. What is a tangent to a parametric equation?

A tangent to a parametric equation is a line that touches the curve defined by the equation at only one point, and has the same slope as the curve at that point.

## 2. How do you find the slope of a tangent to a parametric equation?

The slope of a tangent to a parametric equation can be found by taking the derivative of the equation with respect to the parameter and evaluating it at the specific point of interest.

## 3. Can a parametric equation have multiple tangents at the same point?

Yes, a parametric equation can have multiple tangents at the same point, as long as the curve defined by the equation has a sharp turn or change in direction at that point.

## 4. Are there any specific conditions for a parametric equation to have a tangent?

Yes, for a parametric equation to have a tangent at a specific point, the derivative of the equation with respect to the parameter must exist and be continuous at that point.

## 5. How can tangents to parametric equations be used in real-world applications?

Tangents to parametric equations are commonly used in physics and engineering to analyze the motion of objects along a curve. They can also be used in computer graphics to draw smooth curves and in computer-aided design for creating smooth, curved surfaces.

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