The Scalar (or Dot) Product

If we square both sides of the equation for the magnitude of the vector $\overrightarrow{X} = a_1\underset{\sim}{i} + a_2\underset{\sim}{j}$ , we get

\left|\overrightarrow{X}\right|\cdot\left|\overrightarrow{X}\right| = a_1 \cdot a_1 + a_2 \cdot a_2

This equation is interesting, because we start with a vector and end up with a scalar as the result. If we had a second vector, $\overrightarrow{Y} = b_1\underset{\sim}{i} + b_2\underset{\sim}{j}$ , we could also get a scalar if we calculated

a_1 \cdot b_1 + a_2 \cdot b_2

This is what we call the scalar product of two vectors, sometimes called the dot product because we often write this as $\overrightarrow{X} \cdot \overrightarrow{Y}$ :

\overrightarrow{X} \cdot \overrightarrow{Y} = a_1 \cdot b_1 + a_2 \cdot b_2

The magntiude of a vector can then be related back to the dot product by

\left|\overrightarrow{X}\right|^2 = \overrightarrow{X} \cdot \overrightarrow{X}

We will also see that the dot product can be used to calculate the angle between two vectors $\overrightarrow{X}$ and $\overrightarrow{Y}$ .

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