Vectors are Equal iff Components are Equal
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Theorem
Two vector quantities are equal if and only if 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:
- $\mathbf a = \mathbf b$
$\blacksquare$
Sources
- 1921: C.E. Weatherburn: Elementary Vector Analysis ... (previous) ... (next): Chapter $\text I$. Addition and Subtraction of Vectors. Centroids: Components of a Vector: $6$. Resolution of a vector