The Physical Sense of Vectors

Question

I'm learning vectors now. I can solve difficult problems well enough, but there is something I can't wrap my head around.

AB + BC = AC

If AB = 10 and BC = 9, Would that mean that AC which seems much shorter will have the magnitude of 19?

Answer 1

$AC$ will not have magnitude $19$!

Remember, $AB = B - A$ and $BC = C - B$. Hence,

$AC = AB + BC = (B - A) + (C - B) = C - A$

$C - A$ will be the short vector from $A$ to $C$.

For a simple counter-example that vector magnitudes to not add up like regular numbers, consider $X = (1, 0)$. We know that $\vert X \vert = 1$.

However, $|X + (-X)| = |(1, 0) - (1, 0)| = | (0, 0)| = 0$, whereas

$ |X| + |-X| = |(1, 0) + |(-1, 0)| = 1 + 1 = 2 $

Hence, in general $|A + B| \neq |A| + |B|$.

Answer 2

No. It's not true that "the magnitude of the sum is the sum of the magnitudes."