Vectors are Equal iff Components are Equal

Theorem
Two vector quantities are equal they have the same components.

Proof
Let $\mathbf a$ and $\mathbf b$ be vector quantities.

Then by Vector Quantity can be Expressed as Sum of 3 Non-Coplanar Vectors, $\mathbf a$ and $\mathbf b$ can be expressed uniquely as components.

So if $\mathbf a$ and $\mathbf b$ then the components of $\mathbf a$ are the same as the components of $\mathbf b$

Suppose $\mathbf a$ and $\mathbf b$ have the same components: $\mathbf x$, $\mathbf y$ and $\mathbf z$.

Then by definition:
 * $\mathbf a = \mathbf x + \mathbf y + \mathbf z$

and also:
 * $\mathbf b = \mathbf x + \mathbf y + \mathbf z$

and trivially:
 * $\mathbf a = \mathbf b$