How do I prove the following statements (vectors)?

Question

1. if $\vec{a} = \vec{b}$, then $|a| = |b|$

2. if $|a| = |b|$ then $\vec{a} = \vec{b}$

I believe the second one is false, but I have no idea how to prove this mathematically...

Answer 1

Counterexample in the plane: $\begin{bmatrix}1\\0\end{bmatrix}$ and $\begin{bmatrix}0\\1\end{bmatrix}$ have the same length $1$ but are not equal.

The first is just the statement the norm is a function on vectors.

Answer 2

The two vectors have been stated to be equal.

1. if $\vec{a}=\vec{b}$, ...

A vector requires a magnitude and a direction for its complete description. Since, the vectors have been given equal, they have both direction and magnitude equal. The $|\vec{a}|$ and $|\vec{b}|$ denote the magnitude of the two vectors, by the aforesaid statement, they are equal.

if $|a|=|b|$ then $\vec{a} = \vec{b}$

As Gallegos has stated in the comments above, the second statement can be proved to be false by taking the counter-case: $\vec{b}$ = ${-}\vec{a}$.

Again, the magnitudes of the two vectors is same, but the directions are opposite making $\vec{a} \neq \vec{b}$.