Undergrad Is x{hat} a unit vector and why is theta a vector?

[cgreeleybsu]

I have a physics test tommorow, and my physics professor said this homework was important. However I have been having difficulty interpreting it. Can someone help me with this. Is x{hat} a unit vector. The part that says +x{hat} axis seems to imply that, but then where does the length for r{hat} come from. Why is theta a vector?

Im so confused, I don't know when to ask for help sometimes until its too late, please help

 

[jtbell]

Is x{hat} a unit vector.

Yes. $\hat x$ is the unit vector that points in the +x direction. Similarly, $\hat y$ is the unit vector that points in the +y direction. Note that these are not the same things as the x and y coordinates of a point, which don't bear a "hat".

At a given point, $\hat r$ is the unit vector that points away from the origin, in the direction of increasing r. At that point, $\hat \theta$ is the unit vector perpendicular to $\hat r$ that points in the direction of increasing $\theta$ (i.e. counterclockwise around the origin). Note that these are not the same things as the r and $\theta$ coordinates (polar coordinates) of the point, which don't bear a "hat".

I think it would be very helpful to draw a careful diagram, to scale, that shows a point in the second quadrant, and all four of these unit vectors. Choose a point for which the coordinate $\theta$ is not close to 135° (halfway between the +y and -x axes), but instead something like maybe 120° or 150° (closer to one axis or the the other). This will make it easier to see which trig functions (sin or cos) are involved where.

 

[cgreeleybsu]

Yes. $\hat x$ is the unit vector that points in the +x direction. Similarly, $\hat y$ is the unit vector that points in the +y direction. Note that these are not the same things as the x and y coordinates of a point, which don't bear a "hat".

At a given point, $\hat r$ is the unit vector that points away from the origin, in the direction of increasing r. At that point, $\hat \theta$ is the unit vector perpendicular to $\hat r$ that points in the direction of increasing $\theta$ (i.e. counterclockwise around the origin). Note that these are not the same things as the r and $\theta$ coordinates (polar coordinates) of the point, which don't bear a "hat".

I think it would be very helpful to draw a careful diagram, to scale, that shows a point in the second quadrant, and all four of these unit vectors. Choose a point for which the coordinate $\theta$ is not close to 135° (halfway between the +y and -x axes), but instead something like maybe 120° or 150° (closer to one axis or the the other). This will make it easier to see which trig functions (sin or cos) are involved where.

Thank you for the reply, sorry for the late response, so would I be correct in saying $\hat r$ is the gradient (which will only point out because of the quadrant selected), and $\hat θ$ is the curl at that particular location?

 

[Redbelly98]

Thank you for the reply, sorry for the late response, so would I be correct in saying $\hat r$ is the gradient (which will only point out because of the quadrant selected), and $\hat θ$ is the curl at that particular location?

Hi, I know you've already had your test, but presumably knowing about this would be useful for the final exam or for that matter understanding any material that uses coordinates somehow.

The gradient and curl have to do with fields (i.e. scalar or vector functions in space), not with the coordinates themselves. As jtbell indicated, the "hat" vectors are simply unit vectors that indicate the direction of increase of a particular coordinate variable (x, y, r, theta).

To speak of "the gradient", we have to know what scalar function we are taking the gradient of. Likewise for curl and vector functions.

Hope that helps.