# of Two Vectors

By definition, the scalar product, also known as the "dot product",

of two vectors A and B is

A B = A B cos

A B = A B cos

This "dot product" is commutative,

A B = B A

This "dot product" is distributative,

A (B + C) = A B + A C

A = Ax i + Ay j + Azk

B = Bx i + By j + Bzk

A B = (Ax i + Ay j + Azk) (Bx i + By j + Bzk)

= Ax Bx( i i) + Ax By( i j) + Ax Bz( i k) +

+ Ay Bx( j i) + Ay By( j j) + Ay Bz( j k) +

+ Az Bx( k i) + Az By( k j) + Az Bz( k k)

What are these dot products of the unit vectors, i, j, and k?

From our initial definition of the scalar product, A B = A B cos , we have

 i i = 1 1 cos 0 = 1 i j = 1 1 cos 90o = 0 i k = 1 1 cos 90o = 0 j i = 1 1 cos 90o = 0 j j = 1 1 cos 0 = 1 j k = 1 1 cos 90o = 0 k i = 1 1 cos 90o = 0 k j = 1 1 cos 90o = 0 k k = 1 1 cos 0 = 1

 i i = 1 1 cos 0 = 1 i j = 1 1 cos 90o = 0 i k = 1 1 cos 90o = 0 j i = 1 1 cos 90o = 0 j j = 1 1 cos 0 = 1 j k = 1 1 cos 90o = 0 k i = 1 1 cos 90o = 0 k j = 1 1 cos 90o = 0 k k = 1 1 cos 0 = 1

That means

A B = (Ax i + Ay j + Azk) (Bx i + By j + Bzk)

= Ax Bx( i i) + Ax By( i j) + Ax Bz( i k) +

+ Ay Bx( j i) + Ay By( j j) + Ay Bz( j k) +

+ Az Bx( k i) + Az By( k j) + Az Bz( k k)

A B = Ax Bx( 1) + Ax By( 0) + Ax Bz( 0) +

+ Ay Bx( 0) + Ay By( 1) + Ay Bz( 0) +

+ Az Bx( 0) + Az By( 0) + Az Bz( 1)

= Ax Bx + Ay By + Az Bz

That is,

A B = Ax Bx + Ay By + Az Bz

The definition of the scalar product,

A B = A B cos

means that work is the scalar product of the Force and the displacement,

W = F s

Sometimes it will be easier to evaluate this as

W = F s = F s cos

and sometimes it will be easier to evaluate this as

W = F s = Fx sx + Fy sy + Fz sz

W = F s = Fx x + Fy y + Fz z