Velocity has a magnitude and a direction and thus it is considered a vector.
But from linear algebra perspective, a vector is an element of a vector space.
A set of mathematical objects can be a vector space if they follow some conditions. One of the conditions is that if we add two vectors we must get another vector from the set.
Which set of vectors should I take as the vector space?
If car A has a velocity $\vec{v}$, can we add this velocity to the velocity of car B and get another vector?
is the velocity of the car B in the same vector space?
What is the physical significance of such addition of vectors?
Best Answer
If we want to be mathematically precise, just saying "velocity is a vector" doesn't cut it.
The definition of velocity is as the time derivative of position. In mathematical terms, this means - regardless of whether we think of position as a point in $\mathbb{R}^n$ or a more general manifold where position itself is not a vector - velocities are tangent vectors to curves $x(t)$ in our position space. In general, you can add two tangent vectors at the same point because they are vectors in the same tangent space, but you cannot add "velocity of car A" to "velocity of car B" unless the two cars are currently colliding and hence these two vectors live at the same point.
Adding two velocities at the same point is just a way to express that it is equivalent to say "This thing is moving at $\sqrt{2} \frac{\mathrm{m}}{\mathrm{s}}$ northwest" and "This thing is moving at $1\frac{\mathrm{m}}{\mathrm{s}}$ north and it is moving at $1\frac{\mathrm{m}}{\mathrm{s}}$ west" - the "and" there corresponds to addition.