Dot Product/Examples/Arbitrary Example 1

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Example of Dot Product

Let $\cdot$ denote the dot product operator.


Let:

\(\ds \mathbf A\) \(=\) \(\ds 6 \mathbf i + 4 \mathbf j + 3 \mathbf k\)
\(\ds \mathbf B\) \(=\) \(\ds 2 \mathbf i - 3 \mathbf j - 3 \mathbf k\)

Then:

$\mathbf A \cdot \mathbf B = -9$

Hence the angle between $\mathbf A$ and $\mathbf B$ is approximately $104.2 \degrees$.


Proof

\(\ds \mathbf A \cdot \mathbf B\) \(=\) \(\ds 6 \times 2 + 4 \times \paren {-3} + 3 \times \paren {-3}\)
\(\ds \) \(=\) \(\ds 12 - 12 - 9\)
\(\ds \) \(=\) \(\ds -9\)


We have:

\(\ds \mathbf A \cdot \mathbf A\) \(=\) \(\ds 6^2 + 4^2 + 3^2\)
\(\ds \) \(=\) \(\ds 36 + 16 + 9\)
\(\ds \) \(=\) \(\ds 61\)
\(\ds \leadsto \ \ \) \(\ds \norm {\mathbf A}\) \(=\) \(\ds \sqrt {61}\)
\(\ds \) \(\approx\) \(\ds 7.81\)


\(\ds \mathbf B \cdot \mathbf B\) \(=\) \(\ds 2^2 + \paren {-3}^2 + \paren {-3}^2\)
\(\ds \) \(=\) \(\ds 4 + 9 + 9\)
\(\ds \) \(=\) \(\ds 22\)
\(\ds \leadsto \ \ \) \(\ds \norm {\mathbf B}\) \(=\) \(\ds \sqrt {22}\)
\(\ds \) \(\approx\) \(\ds 4.69\)


Hence:

\(\ds \mathbf A \cdot \mathbf B\) \(=\) \(\ds \norm {\mathbf A} \norm {\mathbf B} \cos \theta\) Definition of Dot Product
\(\ds \leadsto \ \ \) \(\ds -9\) \(=\) \(\ds \sqrt {61} \times \sqrt {22} \cos \theta\)
\(\ds \leadsto \ \ \) \(\ds \cos \theta\) \(\approx\) \(\ds -0.246\)
\(\ds \leadsto \ \ \) \(\ds \theta\) \(\approx\) \(\ds 104.2 \degrees\)

$\blacksquare$


Sources

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