## 3.3 The Dot Product

Given two vectors **v** and **w** whose components are elements of R, with the same
number of components, we define their **dot product,** written as
**vw or
(v, w)** as **the sum of the products of corresponding components: ***.*

**Obvious facts:** the dot product is linear in **v** and in **w**
and is symmetric between them.

We define the **length of v** to be the positive square root of **(v,
v); **the** length of v **is usually denoted by **|v|**.

**Wonderful Fact: the dot product is invariant under rotation of coordinates.**

**Exercises 3.1 Prove this statement.** Solution

As a consequence of this fact, in evaluating** vw**, we can rotate coordinates so that the first basis vector is in the direction
of **v** and the second one is perpendicular to it in the plane of **v** and
**w**.

**Then v will have first two coordinates (|v|, 0) and if the angle between
v and w is ,
w will have (|w|cos,
|w|sin)
as its similarly defined coordinates.**

The dot product v**w
**therefore is** |v||w| cos,
**in this coordinate system (that is, with these basis vectors), and hence
in any coordinate system obtained by rotations from it.

The fact that **the dot product is linear in each of its arguments** is
extremely important and valuable. It means that you can apply **the distributive
law** in either argument to express the dot product of a sum or difference
as the sum or difference of the dot products.

**Example**

**Exercises 3.2 Express the square of the area
of a parallelogram with sides v and w in terms of dot products.** Solution

The dot product of **v** and **w** divided
by the magnitude of **w**, which is **|v|cos**,
is called **the component of v in the direction of w.**

The vector in the **w** direction having magnitude and sign of |**v**|**cos**
is called **the projection of v on w.**

The vector obtained by subtracting the projection of **v** on **w** from
**v** is called **the projection of v perpendicular to w** or normal to
**w**. (By definition this projection has zero component in the direction
of **w**, and is therefore normal to **w**.)

**Exercises:**

**3.3 Express the square of the component of v in the direction of w in terms
of dot products.** Solution

**3.4 Express the component of v perpendicular to w in terms of dot products.**
Solution

**3.5 Write out (v - w)****(v
- w) using the linearity of the dot product in each of its arguments. What famous
law does this establish?** Solution

**3.6 Express the projection of v on w in terms of dot products and
the vector w.** Solution