Scalar Product

### Three Names All the Same

Vectors can be multiplied in two different ways, but an SL student only needs to know about the way called the "scalar product" and the result of the multiplication is always a scalar. The second type is not on the SL syllabus, but is useful in many applications including basic physics such as torque.

Math folk seem to have the bad habit of naming the same thing with multiple names (physics folk seem much better at avoiding this). In this case the scalar product, inner product and dot product all refer to the same vector operation. You need to know all three terms.

### Forumlae

The formula for the scalar product can be written two ways, both of which are in the IB data booklet:

(1)
\begin{align} v \cdot w = |v| |w| cos \theta \end{align}

Where $\theta$ is the angle between the two vectors. It can also be written as:

(2)
\begin{align} v \cdot w = v_1 w_1 + v_2 w_2 + v_3 w_3 \end{align}

When the vectors are defined as:

(3)
\begin{align} v={\begin{pmatrix} v_1\\ v_2\\ v_3 \end{pmatrix}} \: \: \: w=\begin{pmatrix} w_1 \\ w_2 \\ w_3 \end{pmatrix} \end{align}

#### Perpendicular & Parallel

If the two vectors $v$ and $w$ are perpendicular then the scalar product will be zero.

(4)
\begin{align} v \cdot w = 0 \end{align}

If the two vectors $v$ and $w$ are parallel then scalar product of their unit vectors will be 1 (i.e. the cosine of the angle between them will equal one).

(5)
\begin{align} \frac{v}{|v|} \cdot \frac{w}{|w|} = cos \theta = 1 \end{align}

The section(s) below are for those wanting to understand a bit more as to where these formula come from. As usual the IB has not left sufficient time to develop either the need or the background for the concept.

### Starting With Projections

If you'd like a bit more explanation expand the links below:

### Scalar Product from Projections

The projection model is limited to unit vectors. So lets try to make that more useful…