Unit Vectors as a Function of Time?

[YoshiMoshi]

Homework Statement

Say I have a vector F something like

F = c₁(t) x^{^} + c₂(t) y^{^}

were c1 and c2 are some scalar functions of time were you plug in time to into the equation and are given some magnitude.

My question seems to be can we define unit vectors/basis vector as a function of time as well? Something like

F = c₁(t) x^{^}(t) + c₂(t) y^{^}(t)

where the unit vector changes with time?

I have always considered unit vectors as being something with a magnitude of one and one defined direction which remains constant. However can I vary the direction of the unit vector as a function of time?

I remember during simple problems in high school physics were you spin a yo yo in a circle at a certain acceleration etc. and we defined "radially inward" from the yo yo as being positive. From the reference of the yo yo the unit vector in this case never changes direction, it's always directly upwards or directly downwards depending on how you look at it. I depict this in the first attached photo. I think in high school we may have even over looked defining the unit vector in this way.

However could I define my system in a way were I move the frame of reference from the center of the yo yo to the center of the circle? In this case my unit vectors which I am defining span the x y plane partially in a doughnut shape that is always one unit above the yo yo at all times. In this case however my unit vector changes direction with time. At the point the yo yo is at it's top point the unit vector is directly downwards, and when it's at it's lowest point it's in the completely opposite direction. This makes my unit vector a function of time. Is it possible to define the system this way? I would assume it's possible, it would over complicate it, but I'm just wondering if it's possible to define unit vectors as a function of time?

Sorry if I didn't explain my question to well.

Homework Equations

The Attempt at a Solution

 

[Chestermiller]

In cylindrical- and spherical coordinates, the unit vectors depend on the spatial coordinates. So, if you move from one location to another, the unit vectors in the coordinate directions change.

 

[YoshiMoshi]

Ah ok thanks I didn't think of that.

 

[Ray Vickson]

Homework Statement

Say I have a vector F something like

F = c₁(t) x^{^} + c₂(t) y^{^}

were c1 and c2 are some scalar functions of time were you plug in time to into the equation and are given some magnitude.

My question seems to be can we define unit vectors/basis vector as a function of time as well? Something like

F = c₁(t) x^{^}(t) + c₂(t) y^{^}(t)

where the unit vector changes with time?

I have always considered unit vectors as being something with a magnitude of one and one defined direction which remains constant. However can I vary the direction of the unit vector as a function of time?

I remember during simple problems in high school physics were you spin a yo yo in a circle at a certain acceleration etc. and we defined "radially inward" from the yo yo as being positive. From the reference of the yo yo the unit vector in this case never changes direction, it's always directly upwards or directly downwards depending on how you look at it. I depict this in the first attached photo. I think in high school we may have even over looked defining the unit vector in this way.

However could I define my system in a way were I move the frame of reference from the center of the yo yo to the center of the circle? In this case my unit vectors which I am defining span the x y plane partially in a doughnut shape that is always one unit above the yo yo at all times. In this case however my unit vector changes direction with time. At the point the yo yo is at it's top point the unit vector is directly downwards, and when it's at it's lowest point it's in the completely opposite direction. This makes my unit vector a function of time. Is it possible to define the system this way? I would assume it's possible, it would over complicate it, but I'm just wondering if it's possible to define unit vectors as a function of time?

Sorry if I didn't explain my question to well.

View attachment 94920 View attachment 94921

Homework Equations

The Attempt at a Solution

Certainly we can have time-dependent unit vectors, even in a Cartesian coordinate system. Just paint unit vectors on the side of the yo-yo, for example.

 

[HallsofIvy]

A vector has two properties- length and direction. A unit vector has constant length, 1, but its direction can be a function of t.