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Inner Product and its Properties

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Definition 1. The inner product of two vectors and is the product of the absolute values of these vectors and the cosine of the angle between them:

.

Property 1. Theinner product of two vectors is equal to the product of the absolute value of one vector and the projection of the second vector onto the first, i.e.,

.

Property 2. The inner product of two vectors is equal to zero if and only if these vectors are perpendicular.

1. If , then , and the inner product equals .

2. Conversely, suppose that

Property 3. The inner product of vectors is commutative:

Property 4. To multiply an inner product by a number l, it is sufficient to multiply one of the factors by l:

(Without proof.)

Property 5. Inner product is associative:

.

 

and .

The last property allows us to multiply these vectors term by term:

.

Since the unit vectors are mutually perpendicular (), their inner products are zero (by Property 2).

Consider the product of a unit vector with itself:

; ; .

Thus, six of the nine terms are zero, and the remaining three terms are

.

The inner product of vectors is equal to the sum of products of their coordinates.


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