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 kth 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 b-w * Pythagorean orthogonal to the subspace W, is only useful in an inner product space, even though its statement only involves norms.