Linear System and Vector equation

[roben19]

\[\begin{array}{l}
\mathop a\limits^ \to = ({a_1},{a_2},{a_3})\\
\overrightarrow b = ({b_1},{b_2},{b_3})\\
\mathop c\limits^ \to = ({c_1},{c_2},{c_3})\\
\mathop d\limits^ \to = ({d_1},{d_2},{d_3})\\
\left\{ \begin{array}{l}
{a_1} \cdot x + {b_1} \cdot y + {c_1} \cdot z = {d_1}\\
{a_2} \cdot x + {b_2} \cdot y + {c_2} \cdot z = {d_2}\\
{a_3} \cdot x + {b_3} \cdot y + {c_3} \cdot z = {d_3}
\end{array} \right.\\
Convert{\rm{ to vector system and solve this to x}}{\rm{.}}\\
\\
\mathop a\limits^ \to \cdot x + \mathop b\limits^ \to \cdot y + \mathop c\limits^ \to \cdot z = \mathop d\limits^ \to ,\\
Solve{\rm{ this to x with multiply }}\mathop {\rm{b}}\limits^ \to {\rm{x}}\mathop {\rm{c}}\limits^ \to (productcross)\\
We{\rm{ have }}\\
\mathop {{\rm{(b}}}\limits^ \to {\rm{x}}\mathop {\rm{c}}\limits^ \to ) \cdot \mathop a\limits^ \to \cdot x + \mathop {{\rm{(b}}}\limits^ \to {\rm{x}}\mathop {\rm{c}}\limits^ \to ) \cdot \mathop b\limits^ \to \cdot y + \mathop {{\rm{(b}}}\limits^ \to {\rm{x}}\mathop {\rm{c}}\limits^ \to ) \cdot \mathop c\limits^ \to \cdot z = \mathop {{\rm{(b}}}\limits^ \to {\rm{x}}\mathop {\rm{c}}\limits^ \to ) \cdot \mathop d\limits^ \to \\
Second{\rm{ and third terms are zero(normally)}}\\
{\rm{Then}}\\
\mathop {{\rm{(b}}}\limits^ \to {\rm{x}}\mathop {\rm{c}}\limits^ \to ) \cdot \mathop a\limits^ \to \cdot x = = \mathop {{\rm{(b}}}\limits^ \to {\rm{x}}\mathop {\rm{c}}\limits^ \to ) \cdot \mathop d\limits^ \to \\

\end{array}\]

[Guest]

This is just Cramer's rule:
https://en.wikipedia.org/wiki/Cramer's_rule
with the determinants replaced by their equivalent vector triple product versions.

You may find the section on the geometric interpretation interesting:
https://en.wikipedia.org/wiki/Cramer's_ ... rpretation

Hope this helped,

R. Baber.

[Guest]

Thanks!