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Dot Product Laws for Dot Product Scalar Projection Vector Projection
Cross Product Laws for Cross Product Triple Product  

 

 

 Vector Multiplication

                     Dot Product

 

Given two vectors "A" and "B", the dot product is defined as the product of the magnitude of A and the magnitude of B and the cosine of the smaller angle between them. The dot of scalar product of A and B is a scalar.

 

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The following laws are valid for Dot Product:

 

1.      A B = B A

 

2.      A (B + C) = A B + A C

 

3.      m(A B) = (mA) B

                     = A (mB)

                    = (A B)m

 

4.      i i = j j = k k = 1

          i j = j k = k i = 0

 

 

5.      A A  = A2

          A A = (Axax + Ayay + Azaz) (Axax + Ayay + Azaz)

 

          A A = (Ax2 ax ax  + Ax Ay ax ay  + Ax Az ax az

     Ay Ax ay ax   + Ay2 ay a+ Ay Az ay az   + 

        Az Ax az a+ Az Ay az ay   + Az2 az az)

 

          |A|2 = {(Ax 2 + Ay 2 + Az 2)}2 = Ax 2 + Ay 2 + Az 2

 

 

6.      If A = Ax i + Ay j + Az k and

             B = Bx i + By j + Bz k

          Then: A B = Ax Bx + Ay By + Az Bz

 

7.      If A B = 0, and A and B are not null vectors, then A and B are perpendicular.

 

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                    Scalar Projection

 

-Scalar component of a vector in the direction of another vector

-The length of projection of a vector on another vector.

 

 

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                    Vector Projection

 

-is the vector component of a vector in the direction of another vector.

 

 

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                    Cross Product

 

The cross product of two vectors say "A" and "B" is by definition:

 

 

where q is the smaller angle between "A" and "B" and an is a unit vector normal to the plane determined by "A" and "B" when they are drawn from a common point. The direction of an is the same as the direction of a right-hand screw where "A" is turned towards "B".

 

 

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The following laws are valid for Cross Product:

 

1.      A x B = -B x A

 

2.      A x (B + C) = A x B + A x C

 

3.      m(A x B) = mA x B

                      = A x mB

                      = (A x B)m

 

4.      If A = Ax i + Ay j + Az k and

             B = Bx i + By j + Bz k

 

 

Expansion by First Row:

 

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Triple Products:

 

1.      (A B)C A(B C)

 

2.      A (B x C) = B (C x A)

                             =C (A x B)

 

         

 

3.      A x (B x C) (A x B) x C

 

4.      A x (B x C) = (A C) B - (A B) C

          (A x B) x C = (A C) B - (B C) A

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