When studying vectors, we all learn the *scalar or dot product*:

\[\begin{equation} \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}||\mathbf{b}| \cos \theta \label{eq:(1)} \end{equation}\]

But what does this mysterious formula actually mean?

### Geometric meaning

The first thing to note is that the angle *θ* is the angle between the vectors **a** and **b**.

Let’s say that vector **a **joins the two points A and B, and that the vector **b** joins the two points A and C. On a diagram, this looks like:

We can see that the |**a**| is the length of the line AB. So, |**a**|cos*θ* gives the length of the *projection* of the line AB on the line AC, which is the line segment AD. Since |**b**| is the length of the line AC, |**a**||**b**|cos*θ* is the length of the line segment AD, multiplied by the length of the line AC.

So, we can say that that dot product gives:

“*The length of the projection of vector a on vector b, multiplied by the length of the vector b*”.

### Link to computation of the dot product

All that is fair and good, but how can we reconcile this with the general method of computation for the dot product which, if **a** = (a_{1}, a_{2}, a_{3}) and **b** = (b_{1}, b_{2}, b_{3}), is:

\[\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3 \label{2} \]

To understand how this method of computation can still give us “the length of the projection of vector **a** on vector **b**, multiplied by the length of the vector **b**”, let us consider the case where the vector **b** is a vector pointing along the x-axis, i.e. **b** = (b_{1}, 0, 0).

In this case, the length of the projection of vector **a** onto the vector **b**, is just the x co-ordinate of **a**, which is a_{1}. Multiplying this by the length of the vector **b**, simply gives us: a_{1}b_{1} – this is the first term of our dot product!

Since we can consider any general vector **b** as being made up of 3 vectors pointing in the x, y, and z-axis, what we are doing then is just projecting the vector **a** onto the component vectors of **b** and adding it up. This naturally gives us the length of the projection of **a** onto **b** multiplied by the length of **b**.

### Conclusion

Vectors are a new and unique concept introduced to students in the A-level curriculum. Dot product, and their link to projecting one vector onto another, is a key element of how vectors behave. So, you need to understand it to start using vectors to solve more complicated problems.

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