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Ok, so I was wondering if I could get a clear explanation of why we solve relative motion problems the way we do?

so I'm given the problem where I have 2 cars, car A and Car B. Now car A is traveling at 24km/hr relative to the earth, and car B is traveling 13km/hr relative to the hr. Now everywhere I looked it tells me, to get the velocity of car A relative to Car B, you essentially subtract 13 from 24 and you have your answer. Now, where I'm confused is how the operation of subtracting the 2 vectors gives us the velocity of A with respect to B?. Essentially the explanation given from my teacher is, "Do steps 1,2 and 3 and you have your answer", however, i want to know why we do what we do to get the answer, if that makes sense.

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Suppose you were sitting in car $A$ then the velocity of Car $B$ that you perceive will be the relative velocity of the car $B$ relative to you (or car $A$).

For example, the velocities of the cars with respect to the earth are,

$\vec{V_a}$ = 24 $\frac{km}{hr}\hat{i}\space and\space \vec{V_b}$ = 13 $\frac{km}{hr}\hat{i}$

Then to obtain the velocity of car $B$ relative to car $A$, imagine what velocity car $B$ will you perceive if you were sitting in car $A$.

In such questions, the observers think of themselves to be at rest.

$Velocity\space of\space B\space with\space respect\space to\space A = \vec{V_{ba}}$ = $\vec{V_b}-\vec{V_a}$

$\vec{V_{ba}}=13\frac{km}{hr}\hat{i}-24\frac{km}{hr}\hat{i}$

$\vec{V_{ba}}=11\frac{km}{hr}\hat{-i}$ (Notice the -ve sign here)

The negative sign indicates that the car $B$ (wrt to you) will come backward (as you think of yourself at rest).

This means that when sitting in the car $A$, you will perceive that the car $B$ is coming towards you with the speed $11\frac{km}{hr}$.

Now if we want to find the velocity of $A$ relative to $B$, sit in the car $B$ and the velocity of car $A$ that you now perceive will be the velocity of car $A$ relative to you.

$\vec{V_{ab}}=11\frac{km}{hr}\hat{i}$

Here, we get a positive value, which means that you (sitting in car B) will perceive the car $A$ come towards you with speed $11\frac{km}{hr}$.

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