Vector Kinematics: Decomposition and Fundamental Properties

1. The Invariance of Vectors and the Role of a Basis

A core principle in physics is that physical reality is independent of the coordinate system we use to describe it. A particle's motion exists whether we observe it or not. The position vector $\vec{r}$ is a geometric object—an arrow in space pointing from an origin to the particle. Its direction and length are absolute.

What changes with the coordinate system are the components of the vector. A basis is a set of linearly independent vectors that span a space. Any vector can be uniquely expressed as a linear combination of these basis vectors. The coefficients of this combination are the vector's components in that basis.

For a deeper dive into the mathematical structure of vectors, bases, and components, please review the following text. Your strong mathematical background will allow you to appreciate the axiomatic approach presented here.

The key takeaway is that while a vector $\vec{v}$ is invariant, its representation $(v_1,v_2,v_3)$ depends entirely on the chosen basis $({\vec{e}}_1,{\vec{e}}_2,{\vec{e}}_3)$.

2. The Calculus of Moving Bases

In a Cartesian system, the basis vectors $\hat{i},\hat{j},\hat{k}$ are constant in both magnitude and direction. This simplifies differentiation enormously:

But what if our basis vectors are not constant? For a coordinate system that moves or rotates, the basis vectors ${\vec{e}}_i(t)$ change with time. To find velocity or acceleration, we must use the product rule:

The central challenge is to find the time derivatives of the basis vectors, $\frac{d{\vec{e}}_i}{dt}$. Let's investigate this through two powerful, non-Cartesian coordinate systems.

3. Polar Coordinates

For motion with rotational symmetry, polar coordinates are often a more natural choice than Cartesian ones. In 2D, a point is described by a radial distance $r$ from the origin and an angle $\theta$ from a reference axis.

The basis vectors are:

• $\hat{r}$: A unit vector pointing radially outward from the origin.
• $\hat{\theta }$: A unit vector perpendicular to $\hat{r}$, pointing in the direction of increasing $\theta$.

Crucially, as a particle moves, its angle $\theta$ changes, and therefore the directions of $\hat{r}$ and $\hat{\theta }$ also change. They are functions of time.

To find their time derivatives, we first express them in the fixed Cartesian basis, a process you will see in the following video. The derivation is a perfect example of proving the fundamental properties of a vector system.

Let's summarize the results of the derivation.

Position:
$\vec{r}=r\hat{r}$

Velocity:
Applying the product rule, $\vec{v}=\frac{d}{dt}(r\hat{r})$, we get:

Here, $\dot{r}$ is the radial velocity (rate of change of distance from the origin), and $r\dot{\theta }$ is the transverse velocity.

Acceleration:
Differentiating the velocity vector is more involved, requiring the product rule on both terms. The result is:

This expression is rich with physical meaning:

• $\ddot{r}\hat{r}$: Linear acceleration in the radial direction.
• $-r{\dot{\theta }}^2\hat{r}$: Centripetal acceleration, directed inward. Note that $\dot{\theta }$ is the angular velocity $\omega$ and $v_t=r\omega$, so this is $-r{\omega }^2\hat{r}=-\frac{v_t^2}{r}\hat{r}$.
• $r\ddot{\theta }\hat{\theta }$: Tangential acceleration due to change in angular velocity. Note that $\ddot{\theta }$ is the angular acceleration $\alpha$, so this is $r\alpha \hat{\theta }$.
• $2\dot{r}\dot{\theta }\hat{\theta }$: The Coriolis acceleration, an interesting term that arises from the interaction between radial motion and rotation.

4. Normal-Tangential (Intrinsic) Coordinates

What if we want a coordinate system that is tied to the path of the particle itself, regardless of any external origin? This is the idea behind the Normal-Tangential or intrinsic coordinate system.

The basis vectors are defined by the geometry of the trajectory:

• Tangential vector (${\hat{u}}_t$): A unit vector tangent to the path, pointing in the direction of velocity.
• Normal vector (${\hat{u}}_n$): A unit vector perpendicular to ${\hat{u}}_t$, pointing toward the instantaneous center of curvature.
• Binormal vector (${\hat{u}}_b$): ${\hat{u}}_b={\hat{u}}_t\times {\hat{u}}_n$, completing a right-handed orthonormal basis.

Let's derive the acceleration in this system.

Velocity:
By definition, the velocity vector is always tangent to the path. Its magnitude is the speed, $v$.

Acceleration:
We differentiate the velocity using the product rule:

This equation already gives us a profound insight. Acceleration has two components:

1. Tangential Acceleration ($a_t=\frac{dv}{dt}$): This component is parallel to the velocity and is responsible for changes in the particle's speed.
2. Normal Acceleration ($a_n$): This component arises from the change in the direction of the velocity vector ($\frac{d{\hat{u}}_t}{dt}$).

To find the form of $a_n$, we must prove a fundamental property. Since ${\hat{u}}_t$ is a unit vector, ${\hat{u}}_t\cdot {\hat{u}}_t=1$. Differentiating with respect to time:

This proves that the vector $\frac{d{\hat{u}}_t}{dt}$ is always orthogonal to ${\hat{u}}_t$. It therefore must lie along the normal direction, ${\hat{u}}_n$.

The following video shows the full derivation for the magnitude of this term.

As derived, $\frac{d{\hat{u}}_t}{dt}=\frac{v}{\rho }{\hat{u}}_n$, where $\rho$ is the instantaneous radius of curvature. Substituting this back gives the complete acceleration vector:

This elegant result cleanly separates the two effects of acceleration: changing speed (tangential component) and changing direction (normal/centripetal component).

5. A Unifying Principle: The Vector Derivative in Rotating Systems

We have seen two examples of moving coordinate systems. There is a single, powerful mathematical theorem that governs them all. For any vector $\vec{A}$ observed from a frame that is rotating with angular velocity $\vec{\omega }$ relative to a fixed frame, its time derivative in the fixed frame is:

This is the transport theorem. It states that the total change of a vector is the sum of its change as seen by the rotating observer, plus the change due to the rotation of the frame itself.

Let's apply this to one of our basis vectors, ${\vec{e}}_i$. Since the basis vectors are fixed in the rotating frame, their derivative in that frame is zero: ${\left(\frac{d{\vec{e}}_i}{dt}\right)}_{\text{rotating}}=\vec{0}$. The theorem then simplifies beautifully:

This single equation is the fundamental property you set out to prove. It's the engine that drives the derivations for any rotating coordinate system. You can explore this concept briefly in the following resource.

Conclusion

In this lesson, we have moved from the static description of vectors to the dynamic calculus of vectors in moving coordinate systems. You have seen that a deep understanding of kinematics requires mastering how to decompose motion and its derivatives in coordinate systems tailored to the problem.

Key Takeaways:

• Physical vectors are invariant, but their components depend on the chosen basis.
• In non-Cartesian systems (like polar or normal-tangential), the basis vectors themselves can change with time.
• Calculating velocity and acceleration in these systems requires using the product rule and finding the time derivatives of the basis vectors.
• We rigorously proved the forms for velocity and acceleration in polar and normal-tangential coordinates, revealing the physical meaning of each term (e.g., centripetal, tangential, Coriolis).
• The fundamental property governing the change of any basis vector in a rotating frame is $\frac{d{\vec{e}}_i}{dt}=\vec{\omega }\times {\vec{e}}_i$.

We have now established the vector algebra and calculus tools essential for describing motion. In the next lesson, we will focus on the definition of velocity itself, bridging the gap between the discrete concept of average velocity and the continuous, instantaneous velocity using the language of limits, and exploring its powerful graphical interpretation.