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

In summary, the curve is defined as x-y^2 = 0, z= x and the goal is to find the tangent line to the curve at the point (1,1,1) in R3. There are infinitely many tangent lines and to specify a line in R3, one can use the form (x1,y1,z1) + t(a,b,c). To write the curve (x,y,z) as a function of a single variable, t is chosen to be x and the vector function is r(t)= (t) i + (sqrt(t)) j + (t) k. To find the slope at t=1 and plug into a vector equation for a line in R3.
  • #1
autre
117
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?
 
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  • #2
There are infinitely many such tangent lines.
 
  • #3
Oh, right, I meant at the point (1,1,1).
 
  • #4
How do you write the "equation" of a line in R3?

Actually, how do you specify a line in R3?
 
  • #5
something like (x1,y1,z1) + t(a,b,c)?
 
  • #6
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'.
 
  • #7
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.
 

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

What is a tangent line to a curve in R3?

A tangent line to a curve in R3 is a line that touches the curve at a specific point and has the same slope as the curve at that point.

How is the slope of a tangent line to a curve in R3 calculated?

The slope of a tangent line to a curve in R3 is calculated using the partial derivatives of the curve's equations with respect to each of the three variables (x, y, and z) at the given point.

What is the significance of a tangent line to a curve in R3?

A tangent line to a curve in R3 can help us understand the behavior of the curve at a specific point. It can also be used to approximate the curve at that point.

Can a curve in R3 have multiple tangent lines at a single point?

No, a curve in R3 can have only one tangent line at a single point. This is because the tangent line represents the best linear approximation of the curve at that point.

How is the tangent line to a curve in R3 different from the tangent plane?

The tangent line to a curve in R3 is a one-dimensional object, while the tangent plane to a surface in R3 is a two-dimensional object. The tangent line represents the slope of the curve at a specific point, while the tangent plane represents the slope in all directions at a specific point on a surface.

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