How to Find the Tangent Line to a Curve in R3 Using Vector Functions?

Click For Summary

Homework Help Overview

The problem involves finding the tangent line to a curve defined by the equations x - y^2 = 0 and z = x in three-dimensional space (R3). The original poster seeks clarification on how to determine the slope of the tangent line at a specific point on the curve.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the nature of tangent lines in R3 and the representation of lines using vector functions. There is a focus on how to express the curve as a function of a single variable and the implications of choosing that variable.

Discussion Status

The discussion is actively exploring different representations of the curve and the corresponding tangent line. Some participants have suggested methods for expressing the curve in terms of a single variable and are considering how to derive the slope at a specific point.

Contextual Notes

There is an emphasis on the point (1,1,1) for finding the tangent line, and participants are questioning how to formulate the line's equation in R3. The discussion includes considerations of multiple tangent lines and the representation of curves in vector form.

autre
Messages
116
Reaction score
0
So the curve is defined as x-y^2 = 0, z= x . I'm supposed to find the tangent line to the curve. How do I find a slope in R3?
 
Physics news on Phys.org
There are infinitely many such tangent lines.
 
Oh, right, I meant at the point (1,1,1).
 
How do you write the "equation" of a line in R3?

Actually, how do you specify a line in R3?
 
something like (x1,y1,z1) + t(a,b,c)?
 
autre said:
something like (x1,y1,z1) + t(a,b,c)?

Ok, can you write the curve (x,y,z) as a function of a single variable? Call that variable 't'.
 
Try letting t=x. Then, x=z=t, y=sqrt(t). So, a vector function for your curve should be r(t)= (t) i + (sqrt(t)) j + (t) k (let i, j, k be unit vectors).

Your point occurs at t=?

Find the slope of the vector function at that t, and plug into a vector equation for a line.
 

Similar threads

Can a Function Have Two Different Tangent Lines at the Same Point?
  • · Replies 2 ·
Replies
2
Views
2K
Write the tangent lines to the curve
  • · Replies 1 ·
Replies
1
Views
1K
Lagrange multipliers understanding
  • · Replies 8 ·
Replies
8
Views
2K
Quadratics Having a Common Tangent
  • · Replies 1 ·
Replies
1
Views
1K
Find the equation of the tangent plane and normal to a surface
  • · Replies 1 ·
Replies
1
Views
2K
Slope of tangent line to curves cut from surface
  • · Replies 3 ·
Replies
3
Views
2K
Concavity and Tangent Functions
  • · Replies 2 ·
Replies
2
Views
1K
Find the equation of a tangent line to y = x^2?
  • · Replies 11 ·
Replies
11
Views
5K
Verify Unit length to y-axis from Tractrix Curve
  • · Replies 1 ·
Replies
1
Views
2K
Find eqn for "normal" line to the tangent- folium
  • · Replies 6 ·
Replies
6
Views
2K