Question

Find the equation of the line tangent to the curve at the indicated \(t\) -value using the unit tangent vector. $$ \vec{r}(t)=\left\langle 2 t^{2}, t^{2}-t\right\rangle, \quad t=1 $$

Short answer

The tangent line is described by the parametric equations \( x = 2 + \frac{4}{\sqrt{17}}t' \) and \( y = \frac{1}{\sqrt{17}}t' \).

Step-by-step solution

Find the Derivative of the Vector Function

First, find the derivative of the vector function \( \vec{r}(t) = \langle 2t^2, t^2 - t \rangle \). Compute \( \vec{r}'(t) \) by differentiating each component with respect to \( t \):\[ \vec{r}'(t) = \langle \frac{d}{dt}(2t^2), \frac{d}{dt}(t^2 - t) \rangle = \langle 4t, 2t - 1 \rangle \].

Evaluate the Derivative at the Given Parameter

Evaluate the derivative \( \vec{r}'(t) \) at \( t = 1 \):\[ \vec{r}'(1) = \langle 4 \times 1, 2 \times 1 - 1 \rangle = \langle 4, 1 \rangle \].This vector is tangent to the curve at \( t = 1 \).

Compute the Magnitude of the Tangent Vector

Find the magnitude of the tangent vector \( \langle 4, 1 \rangle \) to convert it to a unit vector:\[ \| \vec{r}'(1) \| = \sqrt{4^2 + 1^2} = \sqrt{17} \].

Find the Unit Tangent Vector

Divide the tangent vector components by its magnitude to find the unit tangent vector:\[ \hat{T}(1) = \left\langle \frac{4}{\sqrt{17}}, \frac{1}{\sqrt{17}} \right\rangle \].

Find the Point on the Curve at \( t = 1 \)

Substitute \( t = 1 \) into the original vector function \( \vec{r}(t) \) to find the point on the curve:\[ \vec{r}(1) = \langle 2(1)^2, 1^2 - 1 \rangle = \langle 2, 0 \rangle \].This gives the point \( (2, 0) \).

Write the Equation of the Tangent Line

Using the point \( (2, 0) \) and the direction \( \hat{T}(1) = \left\langle \frac{4}{\sqrt{17}}, \frac{1}{\sqrt{17}} \right\rangle \), write the parametric equations of the tangent line:\[ x = 2 + \frac{4}{\sqrt{17}}t' \]\[ y = 0 + \frac{1}{\sqrt{17}}t' \]These equations represent the line tangent to the curve at \( t = 1 \).

Key concepts

Unit Tangent Vector

The unit tangent vector is a crucial concept in vector calculus, especially when studying curves in two or three dimensions. When we work with curves defined by a vector function, a tangent vector points in the direction the curve is heading at any given point. However, tangent vectors can vary greatly in magnitude. That's where unit tangent vectors come in. They take the direction of the tangent vector but normalize it, so it always has a magnitude of 1.

To find the unit tangent vector, we first determine the derivative of the vector function, which gives us the tangent vector. Then, we compute its magnitude—how long the vector is—and divide the original components by this magnitude. This process provides a unit tangent vector with the same direction but a standardized length. This normalization is vital for applications, such as determining angles between curves or analyzing motion along the curve, without worrying about varying speeds or distances.

Vector Calculus

Vector calculus is an extension of calculus that deals with vector fields rather than just scalar fields. In many cases, it's applied to functions that include multiple variables and can represent things like physical quantities in space. Working with vector functions allows us to explore properties like motion, fluid flow, and electromagnetic fields through derivatives and integrals.

One of the key operations in vector calculus is taking the derivative of a vector function. This extension of the calculus derivative measures how the vector function changes at a particular point—often what we refer to as a tangent vector. These derivatives help us determine rates of change and directions, essential in exploring dynamics in physics, engineering, and beyond.

Moreover, vector calculus introduces operations like gradient, divergence, and curl, essential for understanding complex systems. Employing these tools, we can analyze how fields behave over time and space, making vector calculus a backbone of many scientific and mathematical disciplines. It emphasizes a thorough understanding of both the quantitative and directional changes within a multi-dimensional system.

Tangent Vector

The tangent vector is one of the fundamental elements when analyzing curves within vector calculus. It represents the direction in which a curve progresses at a specific point. Derived from the derivative of the curve's vector function, the tangent vector gives insight into the trajectory and speed at that particular instance.

The process of finding a tangent vector involves differentiating the vector function with respect to its parameter. This derivative yields another vector that touches the curve at a given point, illustrating how steep or flat the curve is. The tangent vector plays a crucial role in several applications, such as in determining the path of a moving object or establishing the slope of a curve.

However, its versatility does not stop there. Tangent vectors are also essential in forming equations of tangent lines, which approximate the curve locally. This local linear approximation simplifies complex analyses by reducing the curve to a straight line at a particular point, revealing the curve's local behavior without dealing with its entire complexity. Understanding tangent vectors thus opens up a wide range of possibilities for analyzing and simplifying real-world problems.