Whats the velocity vector?

Two friends can race at the exact same speed, yet one may be accelerating while the other is not. Why? The key lies in the velocity vector.

This idea is mathematically subtle, but it can be captured with the right intuition.

Speed is a scalar quantity. It tells you how fast something moves, but not where it is going. Velocity, on the other hand, is a vector. It has both a magnitude, which is the speed, and a direction.

Here is an analogy.

If I tell you the speed of the ball after Cristiano Ronaldo shoots a penalty, is that enough to know whether he scored? Of course not. The direction of the shot is just as important. Imagine he breaks the Guinness World Record for shot speed, the fastest ball ever recorded, but instead of shooting toward the opponent’s goal, he turns around and shoots toward his own goal. No matter how fast the ball is moving, it is clearly not a goal.

So speed alone is not enough information to describe or predict motion. This is why physics uses vector quantities like velocity.

Two velocity vectors can have the same length, meaning the same speed, but point in different directions. This means the velocities are different.

Acceleration as changing velocity

Physicists define acceleration as the instantaneous change in velocity.

There are two ways velocity can change.

First, a change in speed.
If you press the gas pedal in a car, the car speeds up. The velocity vector gets longer. This matches everyday intuition about acceleration.

Second, a change in direction even when speed stays constant.
Suppose you keep the same speed but take a turn. The length of the velocity vector stays the same, but its direction changes. Since the velocity has changed, you are accelerating even though your speed did not change.

This is the crucial point.
Acceleration does not require a change in speed. A change in direction is enough.

Two friends can run at the same speed, but if one is running in a straight line while the other is constantly turning, for example running on a curved track, the second runner’s velocity is continuously changing direction. Therefore that runner is accelerating even though their speed is the same.

Speed stayed the same.
Velocity changed.
So acceleration occurred.

Normalizing removes magnitude and keeps direction

In mathematics there is a process called normalization. The idea is simple. You divide a quantity by its own size.

If you divide your weight by your weight you get $1$.
If you divide your height by your height you also get $1$.

The same idea applies to vectors.

If you divide a vector by its magnitude the resulting vector has magnitude $1$. Its size is removed but its direction is preserved.

Consider two particles moving along the same curve.

One particle slows down, stops, and then speeds up again.
The other particle moves smoothly at constant speed.

Their velocity vectors are different at every instant because velocity includes speed. Since their speeds are different, the lengths of their velocity vectors are different.

However, at a given point on the curve there is something they must share.

Even though their speeds differ, they move along the same path. Said differently, they share the same direction of motion.

Velocity encodes both speed and direction. Since they do not share the same speed, they do not share the same velocity.

This leads to a natural question.

Is there a vector that captures only direction and not speed?

The answer is yes.

The unit tangent vector

The vector that captures only the direction of motion is called the unit tangent vector.

It points along the curve in the direction of motion but contains no information about speed.

Consider a ship moving in the ocean.

Its velocity vector tells you how fast it moves and in which direction.
A compass only tells the heading. It ignores speed completely.

Mathematically, the compass plays the same role as the unit tangent vector. It keeps direction and removes magnitude.

To remove magnitude, we normalize the velocity vector. This means dividing the velocity vector by its magnitude, which is its speed.

Take your age. Divide it by your age. You obtain

$$ \frac{\text{age}}{\text{age}} = 1 $$

Now apply the same idea to vectors.

Let $\vec v$ be a velocity vector. Its magnitude is $|\vec v|$.

The normalized vector is

$$ \frac{\vec v}{|\vec v|} $$

Its magnitude is

$$ \left| \frac{\vec v}{|\vec v|} \right| = \frac{|\vec v|}{|\vec v|} = 1 $$

So the resulting vector always has magnitude $1$. Its direction is unchanged. Its size is fixed. This is why it is called a unit vector.

If a particle has velocity vector $\vec v(t)$, then the unit tangent vector is

$$ \vec T(t) = \frac{\vec v(t)}{|\vec v(t)|} $$

This vector captures only the direction of motion.

Two particles moving along the same curve share the same unit tangent vector at each point even if their speeds are different.