The dot product may be introduced as a generalization of the negative reciprocal rule for the slopes of perpendicular lines that remains valid when one of the slopes is zero.

In **configuration 1**, we see two
perpendicular vectors, **v** and **w**. We denote their components by:

By the negative reciprocal rule, which is based upon congruent and similar triangle observations also found the derivation of the rotation formula,

We call the quantity on the left-hand side of the latter equation
the *dot product* of **v** and **w**. We often use the word *orthogonal*
to describe vectors whose dot product is zero.

In this configuration

You should confirm that their dot product does indeed equal zero.

To advance to the next configuration click on play.

In **configuration 2**, we have rotated
both vectors, maintaining the same angle between them, to a horizontal/vertical
position along the coordinate axes.

In this configuration

While the slope of the line containing **w**, ,
is undefined, the dot product

remains valid. The dot product criterion avoids the need for special cases. Higher dimensional generalizations of the dot product have powerful applications in areas as diverse as signal processing, quantum mechanics, and statistics.

In **configuration 3**, we consider
the meaning of the dot product of non-perpendicular vectors.

In this configuration

and the dot product of **v** and **w** is

In **configuration 4**, both **v**
and **w** have been rotated clockwise by 75° or
radians.

In this configuration

and the dot product of **v** and **w** is

.

Just as in the first two configurations when the pairs of vectors were orthogonal, simultaneous rotation of the two vectors has left the dot product invariant.

The reason can be found in the rotation formula.

If the rotation takes to then is rotated to and is rotated to

The dot product of the rotated vectors is:

using the Pythagorean relationship .

In the final **configuration 5**, we
use this invariance to identify the general geometric interpretation of the
dot product.

We have rotated both vectors a further 45° or
radians clockwise, so that **v** is aligned with the positive horizontal
axis and its second component is zero.

In this configuration

and the dot product of **v** and **w** is

.

In general, when **v** and **w** are simultaneously rotated
so that **v** is aligned with the positive horizontal axis, their components
may be written

where

are the lengths of **v** and **w**,
respectively, and where
is the angle from **v** to **w** (which in this configuration is the same
as the angle from the horizontal axis to **w**.) By the rotation formula
and Pythagoras, if **w** is rotated to the positive horizontal axis, its
components would be

.

We can see that the dot product of **v**
and **w** is the product of the lengths of **v** and **w** and the
cosine of the angle between **v** and **w**. The sign of the angle depends
on whether are going *from ***v** to **w** or from **w** to **v**,
but we may unambiguously refer to the cosine of the angle *between ***v**
and **w** since

.

Since the dot product is invariant under simultaneous rotation
of **v** and **w**, the dot product of **v** and **w** must *always*
equal the product of the lengths of **v** and **w **times the cosine of
the angle between **v** and **w**.

**EXERCISES**

Exercise: Show that the numerical components of **v** and
**w** of configuration 4 are

and that the dot product of **v** and **w,**

does indeed equal

.

Hints: For the first part, start from

.

For the second part, square both sides.

Solution: Square both sides of the double angle formula

.

and use the Pythagorean relation

to obtain

where

then use the quadratic forumla

The other root must be

since the sum of the roots is always minus the second coefficient

which explains the choice of sign in the quadratic formula.

Since 75° is complementary to 15° (75+15=90), they are
related by a reflection in *y=x*, so we

interchange

To simplify the expression for the dot product, square both sides

.

Exercise (Advanced): If we denote the length or *norm *of
a vector **v** by

check that

We call these the Pythagorean orthogonality relations. Interpret
them in terms of two right triangles with **v** and **w** forming the
legs, and **v+w** and** ****v-w**
the hypotenii/hypotenuses, respectively.

Exercise: (Advanced): Show that the *scalar cross product
*of **v** and **w**,

is also invariant under rotations and find a geometric interpretation
in terms of the lengths of **v** and **w** and** **the angle between
them.

Exercise (Advanced): Generalize the dot and cross products and their geometric interpretations to three dimensions.

Hint: Show that two vectors in any number of dimensions may be simultaneously rotated by a sequence of coordinate plane rotations to the standard configuration 5, i.e., where all components but the (non-negative) first component of the first vector are zero, and all but the first two components of the second vector are zero.

This can be further generalized for up to *n* vectors to
make all but the first *k* components of a *k*th vector, equal to
zero. In linear algebra, this is called forming the **A=QR** (orthogonal/triangular)
decomposition of the matrix **A** using *Givens' rotations.* It is also
related to the *Gram-Schmidt process*. Both of these may also be interpreted
in terms of *orthogonalizing* a set of vectors with *triangular* coefficients,
instead of *triangularizing* a set of vectors with *orthogonal* coefficients
as we have here.

Exercise (Advanced): Show that any linear transformation/matrix which preserves all lengths must also preserves all angles.

Exercise (Advanced): Show that any norm obeying the parallelogram law arises from an inner product.

Exercise (Advanced): Show that any norm for which all vectors
that are Pythagorean orthogonal to an arbitrary vector form a subspace arises
from an inner product. This shows that the method of least squares, which characterizes
the best approximation of a vector **b** by a vector **w **in a subspace
**W**, as the vector **w
_{*}
**making the error