# Time Derivative of Unit Vectors

1. Jan 23, 2016

### PhysicsKid0123

Quick question (a little rusty on this): Why don't unit vectors in Cartesian Coordinates not change with time? For example, suppose $$\mathbf{r} (t) = x(t) \mathbf{x} + y(t) \mathbf{y} + z(t) \mathbf{z}$$ How exactly do we know that the unit vectors don't change with time?

Or in other words, what is the argument that justifies this expression: $$\frac{d }{dt}\mathbf{x} = 0$$

Last edited: Jan 23, 2016
2. Jan 23, 2016

### vancouver_water

They can change with time. $\mathbb{r(t)} = \sin{t}\mathbb{x} + \cos{t}\mathbb{y}$ is a unit vector that changes with time. If none of the components depend on time, the derivative will be 0. Otherwise the derivative will be another vector.

3. Jan 23, 2016

### PhysicsKid0123

I'm talking about the unit vectors in Cartesian coordinates themselves $$\mathbf{e}_1 = \mathbf{x}, \mathbf{e}_2 = \mathbf{y}, \mathbf{e}_3 = \mathbf{z}$$

4. Jan 23, 2016

### vancouver_water

$\mathbb{x}$ for example can be defined as the vector from the origin to the point $(1,0,0)$. Since the two points are not changing with time, the vector wont change with time either.

5. Jan 23, 2016

### PhysicsKid0123

okay, that's true, now I remember. Thanks.