Consider two vectors **a** and **b**, the angle between them being α. For the sake of simplicity we have represented them in two dimensions in the figure below:

The **dot product or scalar product returns a scalar** (that is, a number) and is given by:

From the expression above we can see that the **dot product of two perpendicular vectors is 0**.

The dot product can be used to find the projection of a vector onto another one. As an example we are going to find the projection of **a** onto **b**.

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The projection of a vector on a line can be found by drawing a line from the end of the vector perpendicular to the line onto which we want to project it. The resulting distance over the line is the projection a_{b}:

Analytically it can be found by using the cosine function:

The direction defined by a vector is given by the unit vector **u**_{b}:

And performing the dot product on **a** and **u**_{b}:

Which is precisely the projection of **a** onto **b**.

The **dot product is commutative**:

From the previous results, it can be deduced:

When both vectors **a** and **b** are expressed in unit vector notation, as shown in the first figure, the dot product is given by:

Finally, we can find the **angle between two vectors** by using the dot product:

The dot product is used in Physics to define the work of a force.

In the animation below ** b’** represents **b** rotated 90^{0}. Since the cosine is the sine complement, the area of the parallelogram that vectors **a** and **b’** span is the absolute value of the dot product **a** · **b**. You can move both vectors **a** and **b** to see their dot product.

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