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 $(a \cdot a+2a \cdot b+b \cdot b) = c \cdot c$
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Yes, but $a\times a=0$, see here. So $a\times b+b\times a=0$, too. And $(a+b)\times (a+b)=0$. – Dietrich Burde Jun 08 '22 at 19:02
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2The OP is asking about the dot product, not the cross product. – cpiegore Jun 08 '22 at 19:43
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You are correct in stating that $a \times a = 0$. However, $a \cdot a = \left| a \right| ^2$ – cpiegore Jun 08 '22 at 20:02
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1Terminologically, we call it a "product" because it is bilinear like this. – GEdgar Jun 08 '22 at 20:55
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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.
