How to prove that vectors are parallel iff their unit vectors are equal?
$$\mathbf{u} \parallel \mathbf{v} \iff \hat{\mathbf{u}} = \hat{\mathbf{v}}$$
A vector can be written as a scalar multiple of its magnitude and unit vector in its direction: $\mathbf{u}=\|\mathbf{u}\| \hat{\mathbf{u}}$. Intuitively, unit vectors convey the direction and any two vectors with the same unit vector must have the same direction. But how to prove it?
I started:
Vector $\mathbf{u}$ is in the direction of nonzero vector $\mathbf{v}$ iff there exists a positive scalar $\lambda$ which scales vector $\mathbf{v}$ to be equal $\mathbf{u}$ (I don't consider antiparallel vectors here):
$$\mathbf{u} \parallel \mathbf{v} \iff \exists \lambda\in \mathbb{R}^+ \, : \, \mathbf{u}= \lambda\mathbf{v}$$
Hence I try to prove
$$\exists \lambda\in \mathbb{R}^+ \, : \, \mathbf{u}= \lambda\mathbf{v} \iff \hat{\mathbf{u}} = \hat{\mathbf{v}}$$
I'm stuck, any hints?
Best Answer
The question should be rephrased as
Prove that for two non zero vectors $u$ and $v$ , $$u=\lambda v \iff \frac {u}{||u||}=\frac {v}{||v||}$$
The proof is straightforward.