Fun with Math!: The Dot Product

Given two vectors u = < u₁, u₂, u₃ > and v = < v₁, v₂, v₃ >, and the angle between them is $\theta$ . The dot product between these two vectors are defined as below.

$\mathbf{u}\cdot \mathbf{v}=u_1v_1+u_2v_2+u_3v_3$
$\mathbf{u}\cdot \mathbf{v}=|\mathbf{u}||\mathbf{v}|\cos\theta$

We can use either one to find the dot product, depending on the available information.

From the second definition, we can modify the equation to become

$\theta=\cos^{-1}\left(\frac{\mathbf{u}\cdot \mathbf{v}}{|\mathbf{u}||\mathbf{v}|} \right )$
in order to find the angle between the two vectors.

Now, move the points A and B below and observe the changes of the calculation. The calculation on top of the graph is the calculation based on the first definition above. The bottom calculation is based on the second definition. Get yourself a calculator and calculate for both. You will find that both gives to the same answer.

Take note that $\textbf{u}=\overrightarrow{OA}$ and $\textbf{v}=\overrightarrow{OB}$ .

From the second definition, we can deduce the following statement

$\mathbf{u}\cdot \mathbf{v}=|\mathbf{u}||\mathbf{v}| \Leftrightarrow \mathbf{u}\perp \mathbf{v}$ because $\cos(90^{\circ})=0$ .

Betty, Created with GeoGebra and Thomas Calculus, Pearson.

Fun with Math!

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Monday, May 4, 2020

The Dot Product

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