Prove that two vectors parallel to the same vector are parallel to each other

by RoRi

April 25, 2016

In $\mathbb{R}^n$ prove that if two vectors are both parallel to a third vector, then they are parallel to each other.

Proof. Let $A$ and $B$ both be parallel to a vector $C$ . By the definition of parallel, this means that there are nonzero scalars $c_1$ and $c_2$ such that

$c_1 A = C \qquad \text{and} \qquad c_2 B = C.$

But, then since $c_1$ and $c_2$ are nonzero we know $c_2^{-1}$ exists and $c_1 c_2^{-1} \neq 0$ . Hence,

$B = c_1 c_2^{-1} A$

and we have that $A$ and $B$ are parallel as well $. \qquad \blacksquare$

Stumbling Robot

Prove that two vectors parallel to the same vector are parallel to each other

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