Velocity of Point Moving on Surface of Sphere is Perpendicular to Radius

Theorem

Let $P$ be a point moving on the surface of a sphere.

The velocity of $P$ is perpendicular to its radius at $P$.

Proof

Let $S$ be a sphere whose center is at $O$.

By definition of a sphere, all the points on the surface of $S$ are the same distance from its center.

Let $\map {\mathbf v} t$ denote the position vector of $P$ with respect to $O$ at time $t$.

Then the magnitude $\norm {\mathbf v}$ of $\mathbf v$ is contstant.

Hence from Dot Product of Constant Magnitude Vector-Valued Function with its Derivative is Zero:

$\map {\mathbf v} t \cdot \dfrac {\d \map {\mathbf v} t} {\d t} = 0$

That is, the dot product of the velocity of $P$ with the radius vector of $P$ is zero.

Hence by Dot Product of Perpendicular Vectors, the velocity of $P$ is perpendicular to its radius at $P$.

$\blacksquare$

Sources

• 1951: B. Hague: An Introduction to Vector Analysis (5th ed.) ... (previous) ... (next): Chapter $\text {III}$: The Differentiation of Vectors: $2$. Differentiation of Sums and Products