# What's the integral of a unit vector?

• MaestroBach
In summary, the conversation discusses how to integrate a unit vector in cylindrical coordinates and its behavior during a line integral. The example is given using the polar unit vector in terms of Cartesian coordinates. It is concluded that the unit vector does not change during the integral and the integral of a constant unit tangential vector along a smooth path results in the displacement from start to finish.
MaestroBach
Homework Statement
Determine the integral of phi hat with respect to phi.
Relevant Equations
N/A
So I'm trying to figure out the integral of phi hat with respect to phi in cylindrical coordinates. My assumption was that the unit vector would just pass through my integral... is that correct? (I reached this point in life without ever thinking about how vectors go through integrals, and google wasn't the best help).

You'll need to define the integral you want to calculate. The polar unit vectors are functions of position, so you can integrate a unit vector along a curve.

Delta2
PeroK said:
You'll need to define the integral you want to calculate. The polar unit vectors are functions of position, so you can integrate a unit vector along a curve.

If I perform a line integral in the direction of my unit vector, then what happens to the unit vector?

MaestroBach said:
If I perform a line integral in the direction of my unit vector, then what happens to the unit vector?

SammyS
Try expressing the unit vector in Cartesian coordinates but expressing its components in terms of φ.

MaestroBach said:
If I perform a line integral in the direction of my unit vector, then what happens to the unit vector?
NOTHING happens to the unit vector. The unit vector in a given direction is a constant vector and doesn't change.

Do you mean you are integrating a vector field with constant length 1? If so, what is the direction given? Is it constant?

HallsofIvy said:
NOTHING happens to the unit vector. The unit vector in a given direction is a constant vector and doesn't change.
No, the direction of ##\hat\theta## changes along the curve.

Since the thread appears to have fizzled out, here's my answer, using my hint in post #5.
##\hat\phi=-\sin(\phi)\hat x+\cos(\phi)\hat y##
Since ##\hat x## and ##\hat y## are constant we can integrate immediately:
##\int\hat\phi.d\phi=[\cos(\phi)\hat x+\sin(\phi)\hat y]=\Delta\cos(\phi)\hat x+\Delta\sin(\phi)\hat y=\Delta \hat r##

Last edited:
lightlightsup and Delta2
I believe something that relates to this thread: If I say that the integral of a constant in magnitude and tangential in direction vector along a closed curve is zero, am I correct?

Delta2 said:
I believe something that relates to this thread: If I say that the integral of a constant in magnitude and tangential in direction vector along a closed curve is zero, am I correct?
And more generally, the integral of the unit tangential vector along a smooth path is the displacement from start to finish.

lightlightsup and Delta2

## 1. What is a unit vector?

A unit vector is a vector with a magnitude of 1 and is commonly used to represent direction in mathematics and physics.

## 2. What is the purpose of taking the integral of a unit vector?

The integral of a unit vector can be used to calculate the displacement of an object in a given direction over a specific time period.

## 3. How do you calculate the integral of a unit vector?

To calculate the integral of a unit vector, you need to first determine the initial position and the final position of the object in the given direction. Then, multiply the magnitude of the vector by the time interval between the two positions.

## 4. Can the integral of a unit vector be negative?

Yes, the integral of a unit vector can be negative if the object moves in the opposite direction of the unit vector. This indicates that the displacement is in the opposite direction of the unit vector.

## 5. What are some real-life applications of the integral of a unit vector?

The integral of a unit vector is commonly used in physics to calculate displacement, velocity, and acceleration of objects in a specific direction. It also has applications in engineering, navigation, and robotics.

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