Vector Proof (1)

Prove that:

\left( \begin{matrix} { a }_{ 1 } \\ { a }_{ 2 } \\ { a }_{ 3 } \end{matrix} \right) \left( \begin{matrix} { b }_{ 1 } \\ { b }_{ 2 } \\ { b }_{ 3 } \end{matrix} \right) ={ a }_{ 1 }{ b }_{ 1 }+{ a }_{ 2 }{ b }_{ 2 }+{ a }_{ 3 }{ b }_{ 3 }\\ \\

Firstly, look at the image below.

vector image

You should know that, if \underline { a } \\ \\ and \underline { b } \\ \\ are perpendicular \underline { a } \cdot \underline { b } =0\\ \\ .

You should also know these rules:

\left| \underline { i }  \right| \left| \underline { i }  \right| ={ \underline { i }  }^{ 2 }=1\cdot 1=1\\ \\ \left| \underline { j }  \right| \left| \underline { j }  \right| ={ \underline { j }  }^{ 2 }=1\cdot 1=1\\ \\ \left| \underline { k }  \right| \left| \underline { k }  \right| ={ \underline { k }  }^{ 2 }=1\cdot 1=1\\ \\

Knowing these rules, we can say that:

vector image 2

*Click on the proof above to see it in full.

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