Vector as Sum of Orthogonal Base Vectors

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Theorem

Let $\mathbf v$ be a vector quantity in ordinary $3$-space.

Let $\mathbf i, \mathbf j, \mathbf k$ be orthonormal base vectors.

Then:

$\mathbf v = \paren {\mathbf v \cdot \mathbf i} \mathbf i + \paren {\mathbf v \cdot \mathbf j} \mathbf j + \paren {\mathbf v \cdot \mathbf k} \mathbf k$


Proof



Sources