True or False? If False, you should come up with a counterexample. If True, can you give a proof?
(a)
The zero vector is the only vector of length 0.
True False
(b)
If \(\norm {\vec {v}-\vec {w}}=0\), then \(\vec {v}=\vec {w}\).
True False
(c)
If \(\vec {v}=-\vec {v}\), then \(\vec {v}=\vec {0}\).
True False
(d)
If \(\norm {\vec {v}}=\norm {\vec {w}}\), then \(\vec {v}=\vec {w}\).
True False
(e)
If \(\norm {\vec {v}}=\norm {\vec {w}}\), then \(\vec {v}=\pm \vec {w}\).
True False
(f)
If \(\vec {v}=t\vec {w}\) for some scalar \(t\), then \(\vec {v}\) and \(\vec {w}\) have the same direction.
True False
(g)
If \(\vec {v}\), \(\vec {w}\) and \(\vec {v}+\vec {w}\) are non-zero, and \(\vec {v}\) is parallel to \(\vec {v}+\vec {w}\), then \(\vec {v}\) and \(\vec {w}\) are also parallel.
True False
(h)
\(\norm {\vec {v}+\vec {w}}=\norm {\vec {v}}+\norm {\vec {w}}\)
True False

Source

[Nicholson] W. Keith Nicholson, Linear Algebra with Applications, Lyryx 2021, Open Edition, Problem 4.1.21.