Let (V, \langle \cdot, \cdot \rangle) an Euclidean vector space
and v, w, u \in V with {\color{blue}\langle v,u \rangle = A}
and {\color{red}\langle v,w \rangle = D}.
Determine the value of the scalar product \langle Cu + B w, v\rangle.
\langle Cu + B w, v\rangle
=
C*A+D*B
Using the bilinear property, we have
\langle Cu + B w, v\rangle =
\langle Cu, v \rangle + \langle Bw, v \rangle =
C \langle u, v \rangle + B \langle w, v \rangle .
Symmetry of the scalars product delivers
C {\color{blue}\langle v,u \rangle}+ B {\color{red}\langle v,w \rangle}.
\langle Cu + B w, v\rangle = negParens(C) \cdot {\color{blue}negParens(A)}
+ negParens(B)\cdot {\color{red}negParens(D)} = C*A+D*B.