Unit Tangent Vector for a Helix
If you find this page helpful and would recommend that I create more pages like this one, please let me know: Email to John Taylor
Find the unit tangent vector for
Equation 1
for
.
How do we proceed?
We need to find the full tangent vector and divide by its length.
How do we find the tangent vector?
We find it by differentiating the equation of the position vector from Equation 1.
Set that up.
Can we apply the differentiation to each component separately?
Yes.
Show that.
Do the differentiation.
We get
Evaluate this at
.
Equation 2
Is this the unit tangent vector?
No.
What do we need to do additionally?
We need to divide by the length:
How do we evaluate the length?
The length equals the square root of the sum of the squares of the components.
Set that up.
Evaluate this.
Equation 3
Combine Equations 2 and 3 to get the unit tangent vector.
We get
for the unit tangent vector.
What is the shape of this curve?
The trig dependence suggests an ellipse.
What about the y-component?
It increases linearly.
What is the net effect?
A helix.
How is it oriented?
Along the y-axis.
Sketch this helix.
Let's plot the point for
. How do we determine its coordinates?
We substitute
into Equation 1.
Do that.
Plot this point on your paper graph.
Note that the coordinates of the point are shown in color.
Now we'll add the position vector for the point at
. How do we determine its components?
Its components are the same as the coordinates of
.
Where do we draw the base of this vector?
At the origin.
Where do we draw the head of this vector?
At the point.
Add the position vector to the diagram.
Note that the components are shown in color.
Now let's add the tangent vector. (The unit tangent vector is too short to show well in this diagram. It will be parallel to the tangent vector, of course.)
Which equation above gives us the components of the tangential vector?
Equation 2.
Where do we draw the base of the tangent vector?
We draw it at the point on the helix.
Draw the components.
Note that the components are shown in color.
Now add the tangent vector to the diagram.
Do the two vectors appear to be perpendicular?
No. They rarely are. The circle and sphere are special cases.
The end. If you found this helpful and would recommend that I create more pages like this one, please let me know: Email to John Taylor