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Problem 49 Hard Difficulty

Figure 16 gives a geometric demonstration of Property 2 of vectors. Use components to give an algebraic proof of this fact for the case $ n = 2 $.

Answer

$$
\therefore a+(b+c)=(a+b)+c
$$

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Video Transcript

for the given problem, we want to demonstrate property to of vectors. So we have a let's say this is vector A. This is gonna be vector B. And this will be a vector C. So when we have a plus B1st, since this is A. And this is B. And their head to tail. We can always rearrange factors to be head details. So we can just skip that first step. So we know that A plus B is going to be this factor right here, and now we have A plus B. So it's gonna be this fact right here. And now we have plus C. So the A plus B plus C where we see this one's also had detail. So then that means that this new vector is going to be right here. So that's A plus B plus C. Now let's do it the other way. What if we had a B plus C first? Well, that's going to be this factor right here. And then we have a plus B plus C. So that's gonna be this vector right here. And as we see, it's the exact same thing