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--- book:chap2:2.7_the_inner_product [2022/04/22 23:46] – jv
+++ book:chap2:2.7_the_inner_product [2024/12/03 23:28] (current) – jv
@@ Line 41: / Line 41: @@
 $\langle \mathbf v \mid \mathbf v \rangle = 0 \;\Leftrightarrow\;  \mathbf v = \mathbf 0$\\
-For a vector space over $\mathbb{C}$ the requirement a) is replaced by
+For a vector space over $\mathbb{C}$ the inner product returns a complex number, and the constant $c$ is a complex number.
+Moreover, the requirement a) is replaced by
 a)  conjugate symmetry:
 $\langle \mathbf v \mid \mathbf w \rangle = \overline{\langle \mathbf w \mid \mathbf v \rangle}$
-and the constant $c$ is a complex number.
+such that $\langle \mathbf v \mid \mathbf v \rangle$ is a real number.
 </WRAP>
@@ Line 109: / Line 110: @@
     c^2 = a^2 + b^2 - 2\,a\,b\,\cos\theta
 \end{align*}
-Let now $a$, $b$, and $c$ be the length of the vectors $\mathbf a$, $\mathbf b$ and $\mathbf c = \mathbf a - \mathbf b$,
+Let now $a$, $b$, and $c$ be the length of the vectors $\mathbf a$, $\mathbf b$ and $\mathbf c = \mathbf b - \mathbf a$,
 as shown in [[#fig_scalarProductCosSetup |Figure 2.13]].
 Then we have
@@ Line 115: / Line 116: @@
     a^2 + b^2 - 2\,a\,b\,\cos\theta
     &= c^2 = \mathbf c \cdot \mathbf c
+      = (\mathbf b - \mathbf a) \cdot  (\mathbf b - \mathbf a)
       = (\mathbf a - \mathbf b) \cdot  (\mathbf a - \mathbf b)
     \\
     &= \mathbf a \cdot \mathbf a - 2\, \mathbf a \cdot \mathbf b + \mathbf b \cdot \mathbf b
       = a^2 + b^2 - 2\, \mathbf a \cdot \mathbf b
-    \\
+    \\[2mm]
     \Rightarrow\quad
-    \mathbf a \cdot \mathbf b = |\mathbf a| \, |\mathbf b| \, \cos\theta
+    \mathbf a \cdot \mathbf b &= |\mathbf a| \, |\mathbf b| \, \cos\theta
     \\
 \end{align*}