Inner Product

Projection Map

, we consider a unit vector

. We define the projection map onto the line containing vector

as follows: For any vector

such that

is the signed distance from the origin to the foot of the perpendicular to the line containing

from the arrowhead of

. The sign is positive/negative if the vector from the origin to the foot of the perpendicular is in the same/opposite direction of

, as shown in the applet below. As illustrated by the applet below,

is in fact a linear transformation. Therefore, it can be represented by an 1 x 2 matrix. Moreover, it can be shown that

and

, where

. Hence, for any

, we have

Inner Product

The definition of projection map can readily be generalized to

or even

i.e. suppose

is a unit vector in

. Then for any vector

, we define the projection map onto the line containing

as follows:

Since the above definition is symmetric in

and

, this suggests that we should extend the definition to any vector

, without the restriction that

is a unit vector: Definition: Given any vectors

, the real number

is called the inner product (or dot product) of

and

, usually denoted by

. Remarks:

If , then we can write , where is the unit vector in the direction of and is the length of the vector . Hence . Moreover, for any nonzero vector , is perpendicular to if and only if , or equivalently, .
The geometric concept of the "angle between two vectors" is encoded in the inner product: Let and be two unit vectors in or , then , where is the angle between the two vectors. Therefore, in , we can define the "angle between two vectors" through inner product.

The following are some basic properties of the inner product: Let

and

be vectors in

, and let

be any real number. Then

and if and only if

Notice that for any vector

(or

), which is the square of the length of the vector. Hence, we can generalize this definition to vectors in

as follows: Definition: The length (or norm) of vector

is the nonnegative real number

defined by

Given any nonzero vector

, we can compute the unit vector

in the direction of

as follows:

For any two vectors

. We can measure the distance between the arrowheads of the two vectors by finding the length of the vector from one arrowhead to another i.e. the distance is

Exercise

Let in .

Find the unit vector .
Let be the line through the origin containing . Using the inner product, find the perpendicular distance from the point to the line .

Suppose are two vectors in .

Prove that . (Hint: write the norms in terms of inner products)
Using (1), prove that . (Note: This is called the polarization identity. Using it, we can define the inner product in terms of the norm)
Using (1), prove the law of cosine for triangles on a plane.

Inner Product

Projection Map

Inner Product

Exercise

Nowe zasoby

Odkryj zasoby

Odkryj tematy