# Parameterized tangent line to a parameterized curve

• Cauchy1789
In summary, the conversation discusses the construction of a tangent line to a parameterized curve in \mathbb{R}^3 and the theorem that allows for its construction using the slope of the derivative and the curve itself. The tangent line is represented by the equation L(s) = a(t) + s * a'(t) and the constant in the equation is necessary to satisfy the condition L(0) = a(t).
Cauchy1789

## Homework Statement

I seem to remember that a parameterized a(t) curve in $$\mathbb{R}^3$$ that one can construct the tangent from the slope of a'(t) and the curve itself.

such that the tangent line L = a(t) + s * a'(t) to a. This is supposedly a straight line in $$\mathbb{R}^3$$.
To make a long question. What theorem allows me to construct the tangent in such a way? confused:

Cauchy1789 said:
I seem to remember that a parameterized a(t) curve in $$\mathbb{R}^3$$ that one can construct the tangent from the slope of a'(t) and the curve itself.

such that the tangent line L = a(t) + s * a'(t) to a. This is supposedly a straight line in $$\mathbb{R}^3$$.
To make a long question. What theorem allows me to construct the tangent in such a way? confused:

Hi Cauchy1789!

It's the theorem that says that the slope of the tangent euqals the derivative …

so, for fixed t, L(s) = s * a'(t) + constant …

and the constant has to be a(t) because L(0) = a(t).

## What is a parameterized curve?

A parameterized curve is a mathematical representation of a curve or path that is defined by a set of equations, with one or more parameters that vary. These parameters can be thought of as variables that determine the shape, position, and orientation of the curve.

## What is a parameterized tangent line?

A parameterized tangent line is a line that touches a parameterized curve at a specific point, following the direction of the curve at that point. It is defined by the equations of the curve and the derivative of those equations with respect to the parameter.

## How is a parameterized tangent line calculated?

The parameterized tangent line can be calculated by finding the slope of the curve at the given point, using the derivative of the curve's equations. This slope is then used to find the equation of the line that passes through the given point and has the same slope.

## Why is the parameterized tangent line useful?

The parameterized tangent line is useful in many applications, such as physics and engineering, as it provides information about the direction and rate of change of a curve at a specific point. It can also be used to approximate the behavior of a curve near the given point.

## Can a parameterized tangent line be vertical?

No, a parameterized tangent line cannot be vertical, as this would mean the slope of the curve at that point is infinite. The slope of a curve at a specific point can only be infinite if the curve has a discontinuity or vertical tangent at that point.

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