Is the Velocity Vector Perpendicular to the Position Vector?

[Tasell]

A point M is located by the vector r(t), which depends on time, but the length of r(t) is constant. Show that the velocity v(t) of M is perpendicular to r(t).

 

[arildno]

Well, what can you say about the quantity:
$$\vec{r}\cdot\vec{r}$$

 

[thomate1]

I would like to begin by clarifying that the vector r(t) represents the position of point M at a given time t. This vector has both magnitude (length) and direction, and it changes with time since the position of point M is dependent on time.

Now, the statement mentions that the length of r(t) is constant, which means that the magnitude of this vector does not change with time. This implies that the direction of r(t) is the only component that varies with time.

To understand the relationship between position and velocity vectors, we need to recall the definition of velocity. Velocity is the rate of change of position with respect to time, or in other words, it is the derivative of position with respect to time.

Therefore, the velocity vector v(t) can be written as the derivative of r(t) with respect to time, which can be expressed as v(t) = dr(t)/dt.

Now, if we consider the magnitude of v(t), it represents the speed at which point M is moving. Since the length of r(t) is constant, the speed of point M is also constant. This means that the magnitude of v(t) is a constant value.

However, the direction of v(t) is determined by the direction of r(t) and how it changes with time. As we know, the derivative of a constant value is equal to zero. Therefore, the direction of v(t) is perpendicular to r(t) since the change in the direction of r(t) is perpendicular to the direction of r(t) itself.

In conclusion, we can say that the velocity vector v(t) of point M is perpendicular to the position vector r(t) since the magnitude of r(t) is constant and the direction of v(t) is determined by the change in the direction of r(t). This is a fundamental concept in physics and is known as the tangent-velocity relationship.