How to Find the Unit Vector for a Parallel Line at a Given Point on tan(x)?

In summary, to find the unit vector parallel to tan(x) at (Pi/4, 1), you can parametrize the path r(t) corresponding to the graph of tan(x) or use the relationship between dy/dx and the y and x components of the unit vector. To find the unit vector normal to tan(x) at (Pi/4, 1), you can use the vector found from the previous step and determine the perpendicular vector with a length of 1.
  • #1
JeffNYC
26
0
A) Find the unit vector parallel to tan(x) at (Pi/4, 1)

B) Find the unit vector normal to tan(x) at (Pi/4, 1)

dy/dx tanx = sec2x and sec2x evaluated at x = Pi/4 is 2. So the slope of the parallel line is 2, but how do I then derive the unit vector?
 
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  • #2
One way is to parametrise the path r(t) corresponding to the graph of tan(x). Then find r'(t). Another way is to note that dy/dx gives you the relationship between the y and x components of the unit vector. Draw out the dy/dx triangle to see why. Now you should be able to find it.

B)Now you've got the vector you should be able to find the unit vector perpendicular to it.
 
  • #3
Defennder,

Can you elaborate on what y' tells me about the y and x components of the unit vector, or point me to the triangle you mentioned?
 
  • #4
dy/dx is the ratio of the y-component of the tangent vector to the x-component. Coupling that with the requirement that the length be 1 let's you determine both. For example, if dy/dt= vy/vx= 2 then vy= 2vx. If then [itex]\sqrt{v_x^2+ v_y^2}= 1[/itex], we have vx2+ vy2= vx2+ 4vx2= 5vx[/sub]2= 5 so [itex]v_x= 1/\sqrt{5}[/itex] and [itex]v_y= 2/\sqrt{5}[/itex].
 

1. What is a unit vector?

A unit vector is a vector with a magnitude of one and is often used to represent a direction in space.

2. How do you find the unit vector?

To find the unit vector, divide each component of the original vector by its magnitude. This will result in a vector with a magnitude of one.

3. Why is the unit vector important in physics and mathematics?

The unit vector is important because it can be used to represent direction and simplify calculations in physics and mathematics. It also allows for easy comparison and analysis of vectors.

4. Can a unit vector have a negative value?

Yes, a unit vector can have negative values. The magnitude of a unit vector is always one, but its direction can be positive or negative depending on the coordinate system being used.

5. How is the unit vector denoted?

The unit vector is denoted by placing a hat (^) over the vector symbol. For example, the unit vector of vector v would be written as v̂.

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