Is it possible to square vector like this?

Question

I'm wondering if squaring using a dot product is correct. For example, let $a,b,c$ be vectors if $(a+b) = c$ then does squaring it mean that $(a+b) \cdot (a+b) = c \cdot c$ i.e does $(a \cdot a+2a \cdot b+b \cdot b) = c \cdot c$

Answer 1

Well, examine it in Cartesian coordinate form.

Because $a + b = c$, we have $$a_i + b_i = c_i \tag{1}\label{1}$$ where subscript $i$ indicates the coordinate component. For 2D, we have $i = 1, 2$ for $x$ and $y$ axes; for 3D, $i = 1, 2, 3$ for $x$, $y$, and $z$ axes; and so on.

For $(a + b) \cdot (a + b) = c \cdot c$ we have $$\sum_i (a_i + b_i)^2 = \sum_i c_i^2 \tag{2}\label{2}$$

If we substitute $\eqref{1}$ into $\eqref{2}$, we see that it is obviously true: both sides become the same expression.