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If I have an equation with two vectors:

$$\pmb{a} = 5\pmb{b}$$

Can we perform dot product on both sides with the same vector like this?

$$\pmb{a·c} = 5\pmb{b·c}$$

If so how could we prove that this a valid operation?

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1 Answer

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Yes, you can dot both sides with the same vector. If $\boldsymbol{a} = 5\boldsymbol{b}$, then by definition of equality, we have $\boldsymbol{a} \cdot \boldsymbol{c} = \left(5\boldsymbol{b}\right) \cdot \boldsymbol{c}$. By the definition of the dot product, it is linear in both inputs (being an inner product on a vector space), so $\left(5\boldsymbol{b}\right) \cdot \boldsymbol{c} = 5\left(\boldsymbol{b} \cdot \boldsymbol{c}\right) = 5\boldsymbol{b} \cdot \boldsymbol{c}$. Thus $\boldsymbol{a} \cdot \boldsymbol{c} = 5\boldsymbol{b} \cdot \boldsymbol{c}$ as desired.

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