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Attributes of scalar (dot) product

In this section, we summarize the properties of dot product as discussed above. Besides, some additional derived attributes are included for reference.

1: Dot product is commutative

This means that the dot product of vectors is not dependent on the sequence of vectors :

a . b = b . a

We must, however, be careful while writing sequence of dot product. For example, writing a sequence involving three vectors like a.b.c is incorrect. For, dot product of any two vectors is a scalar. As dot product is defined for two vectors (not one vector and one scalar), the resulting dot product of a scalar ( a.b ) and that of third vector c has no meaning.

2: Distributive property of dot product :

a . ( b + c ) = a . b + a . c

3: The dot product of a vector with itself is equal to the square of the magnitude of the vector.

a . a = a x a cosθ = a 2 cos 0 ° = a 2

4: The magnitude of dot product of two vectors can be obtained in either of the following manner :

a . b = a b cos θ a . b = a b cos θ = a x ( b cos θ ) = a x component of b along a a . b = a b cos θ = ( a cos θ ) x b = b x component of a along b

The dot product of two vectors is equal to the algebraic product of magnitude of one vector and component of second vector in the direction of first vector.

5: The cosine of the angle between two vectors can be obtained in terms of dot product as :

a . b = a b cosθ

cosθ = a . b a b

6: The condition of two perpendicular vectors in terms of dot product is given by :

a . b = a b cos 90 ° = 0

7: Properties of dot product with respect to unit vectors along the axes of rectangular coordinate system are :

i . i = j . j = k . k = 1 i . j = j . k = k . i = 0

8: Dot product in component form is :

a . b = a x b x + a y b y + a z b z

9: The dot product does not yield to cancellation. For example, if a.b = a.c , then we can not conclude that b = c . Rearranging, we have :

a . b - a . c = 0 a . ( b - c ) = 0

This means that a and ( b - c ) are perpendicular to each other. In turn, this implies that ( b - c ) is not equal to zero (null vector). Hence, b is not equal to c as we would get after cancellation.

We can understand this difference with respect to cancellation more explicitly by working through the problem given here :

Problem : Verify vector equality B = C , if A.B = A.C .

Solution : The given equality of dot products is :

A . B = A . C

We should understand that dot product is not a simple algebraic product of two numbers (read magnitudes). The angle between two vectors plays a role in determining the magnitude of the dot product. Hence, it is entirely possible that vectors B and C are different yet their dot products with common vector A are equal. Let θ 1 and θ 2 be the angles for first and second pairs of dot products. Then,

A . B = A . C

AB cos θ 1 = AC cos θ 2

If θ 1 = θ 2 , then B = C . However, if θ 1 θ 2 , then B C .

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Law of cosine and dot product

Law of cosine relates sides of a triangle with one included angle. We can determine this relationship using property of a dot product. Let three vectors are represented by sides of the triangle such that closing side is the sum of other two vectors. Then applying triangle law of addition :

Cosine law

Cosine law

c = ( a + b )

We know that the dot product of a vector with itself is equal to the square of the magnitude of the vector. Hence,

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Source:  OpenStax, Physics for k-12. OpenStax CNX. Sep 07, 2009 Download for free at http://cnx.org/content/col10322/1.175
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