Vector norm confusion

Question

Given the following vector:

$v = (3,5)$,

I was told that the normalized vector which must have unit length is given by: $v' = (3/\sqrt{34}, 5/\sqrt{34})$

However, why cannot this vector be a normalized vector as well? $ v = (\sqrt{3/8}, \sqrt{5/8})$, which also gives a vector of unit length?

Thanks.

Answer 1

$(\sqrt{3/8},\sqrt{5/8})$ is a normalized vector, but it doesn't have the same direction as $(3,5)$.

The former has the direction of the line $y=\sqrt{\dfrac53} x$,

whereas the latter has the direction of the line $y=\dfrac53x$.

Answer 2

We have that $|v|=\sqrt{34}$ and the

$$\frac{v}{|v|} = \left(\frac3{\sqrt{34}},\frac5{\sqrt{34}}\right)$$

while $w = (\sqrt{3/8}, \sqrt{5/8})$ points in another direction indeed

$$\frac{v_y}{v_x}=\frac 5 3 \neq \frac{w_y}{w_x}=\sqrt {\frac 5 3}$$