I don't even understand what this problem is asking

Question

What does this mean? Looks like everything is equal since there's nothing in the problem that indicates parallel, perpendicular or $0$?

Answer 1

1. the length of vector sum equals to the sum of vector lengths - this means that two vectors parallel

2. whether you add or subtract vector b, the result does not change - this means that b = 0

3. the length of vector sum equals "pythagorean sum" of vector lengths - this means that vectors perpendicular.

Answer 2

1) Triangle inequality says: $|\vec a + \vec b | \le |\vec a| + |\vec b| $. When does equality hold?

2) $\vec a + \vec b = \vec a - \vec b \implies$

$\vec a - \vec a + \vec b = \vec a - \vec a - \vec b \implies$

$\vec b = -\vec b$.

What does that imply?

3) What does $\vec a + \vec b = \vec c$ mean. Well, the naive and intuitive idea is that if you place $\vec b$ and the endpoint of $\vec a$ and view the vector resulting from the origin of $\vec a$ to the endpoint of $\vec b$ you get a third vector, $\vec c$. $\vec a$, $\vec b$ and $\vec c$ form a triangle with sides of lengths $|a|, |b|$ and $|c|$. (Thats why it is called the triangle inequality.)

Keeping that in mind what does $\vec a +\vec b = \vec c$ so that $|a|^2 + |b|^2 = |c|^2$ imply about the triangle formed? What does that say about the vectors?