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Product of Vectors

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Scalar Product

If #~a and #~b are two vectors with an angle &theta. between them, then we define their #~{scalar product} or #~{dot product}, denoted ~#a &dot. #~b is the real number given by:

  • #~a &dot. ~#b _ = _ | ~#a | | ~#b | cos &theta.

The following are fairly obvious:

  • #~b &dot. ~#a _ = _ ~#a &dot. #~b
  • ( λ #~a ) &dot. #~b _ = _ λ ( #~a &dot. #~b )

#~a &dot. ~#b = 0 _ <=> _ ~#a = 0 or #~b = 0 or cos &theta. = 0. _ Cos &theta. = 0 if ~#a and #~b are perpendicular, so we have, for _ #~a != 0, #~b != 0 :

  • #~a &dot. ~#b = 0 _ <=> _ #~a &perp. ~#b

The following also hold for scalar product:

  • #~a &dot. ( ~#b + #~c ) _ = _ ~#a &dot. #~b + #~a &dot. #~c;

Scalar Product by Coordinates

If _ #~a _ = _ ~a_~x #~i + ~a_~y #~j + ~a_~z #~k _ and _ #~b _ = _ ~b_~x #~i + ~b_~y #~j + ~b_~z #~k, _ then

#~a &dot. ~#b _ = _ ( ~a_~x #~i + ~a_~y #~j + ~a_~z #~k ) &dot. ( ~b_~x #~i + ~b_~y #~j + ~b_~z #~k )

_ _ _ _ _ = _ ~a_~x ~b_~x + ~a_~y ~b_~y + ~a_~z ~b_~z

since _ _ #~i &dot. #~i = #~j &dot. #~j = #~k &dot. #~k = 1 and _ _ #~i &dot. #~j = #~j &dot. #~i = #~j &dot. #~k = #~k &dot. #~j = #~k &dot. #~i = #~i &dot. #~k = 0

A facility for calculating dot products of vectors can be found in the MathymaStats suite of programs.

Vector Product

If #~a and #~b are two vectors with an angle &theta. between them, then we define their #~{vector product} or #~{cross product}, denoted ~#a # #~b is the vector for which:

_ _ _ | #~a # ~#b | _ = _ | ~#a | | ~#b | sin &theta.

#~a # ~#b being at right angles to both ~#a and ~#b, such that ~#a, #~b, #~c form a #~{right handed set}, i.e. if the angle &theta. is anti-clockwise from ~#a to #~b, in the plane of ~#a and #~b, then #~a # ~#b is upwards out of the plane, like the direction of a conventional screw (see diagram).

 

We have

  • #~b # ~#a _ = _ - ~#a # #~b
  • ( λ #~a ) # #~b _ = _ λ ( #~a # #~b )

The magnitude _ | ~#a | | ~#b | sin &theta. _ represents the area of the parallelogram with #~a and #~b as two of its sides.

Vector Product by Coordinates

If _ #~a _ = _ ~a_~x #~i + ~a_~y #~j + ~a_~z #~k _ and _ #~b _ = _ ~b_~x #~i + ~b_~y #~j + ~b_~z #~k, _ then

#~a # ~#b _ = _ ( ~a_~x #~i + ~a_~y #~j + ~a_~z #~k ) # ( ~b_~x #~i + ~b_~y #~j + ~b_~z #~k )

_ _ _ _ _ = _ ~a_~x ~b_~y #~k - ~a_~x ~b_~z #~j - ~a_~y ~b_~x #~k ~a_~y ~b_~z #~i + ~a_~z ~b_~x #~j - ~a_~z ~b_~y #~i

_ _ _ _ _ = _ (~a_~y ~b_~z - ~a_~z ~b_~y ) #~i + (~a_~z ~b_~x - ~a_~x ~b_~z) #~j + (~a_~x ~b_~y - ~a_~y ~b_~x ) #~k

since _ _ #~i # #~i = #~j # #~j = #~k # #~k = 0 and _ _ #~i # #~j = #~k, _ #~j # #~i = -#~k, _ #~j # #~k = #~i, _ #~k # #~j = -#~i, _ #~k # #~i = #~j, _ #~i # #~k = -#~j

This can be written

#~a # ~#b _ = _ det{#~i, #~j,#~k/~a_~x,~a_~y,~a_~z/~b_~x,~b_~y,~b_~z}

A facility for calculating vector products of vectors can be found in the MathymaStats suite of programs.

Scalar Triple Product

The quantity ( #~a # #~b ) &dot. #~c is called the #~{scalar triple product} of #~a, #~b, and #~c.

Suppose ~#a, #~b, #~c form a right-handed set i.e. the angle between ~#c and #~a # #~b is acute, &phi. say. Now | #~a # #~b | is the area of the parallelogram with ~#a and #~b as two sides, and _ | #~c | cos &phi. _ is the height of the parallelopied with sides ~#a ~#b and ~#c. So the volume of the parallelopied is:

_ _ ~{base} # ~{height} _ = _ | #~a # #~b | | #~c | cos &phi. _ = _ ( #~a # #~b ) &dot. #~c

Now ~#c, ~#a, ~#b is also a right-handed set, forming the same parallelopied, and so by symmetry:

_ _ ~{volume} _ = _ ( #~a # #~b ) &dot. #~c _ = _ ( #~c # #~a ) &dot. #~b _ = _ ( #~b # #~c ) &dot. #~a

and by commutativity of the scalar product these are also equal to:

_ _ _ _ _ _ = _ #~c &dot. ( #~a # #~b )_ = _ #~b &dot. ( #~c # #~a )_ = _ #~a &dot. ( #~b # #~c )

So the scalar triple product of any even permutation of ~#a, ~#b, and ~#c is the same. We denote this quantity by _ [ #~a, #~b, #~c ].

Since _ #~a # #~b _ = _ - #~b # #~a _ then _ [ #~a, #~b, #~c ] _ = _ - [ #~b, #~a, #~c ] , _ etc.

Vector Triple Product

( #~a # #~b ) # #~c is called a #~{vector triple product} of #~a, #~b, and #~c. We will show that, in general _ ( #~a # #~b ) # #~c _ != _ #~a # ( #~b # #~c )

( #~a # #~b ) # #~c _ is perpendicular to both ( #~a # #~b ) and #~c. In particular, as it is perpendicular to ( #~a # #~b ) then it must lie in the plane of #~a and #~b, i.e _ ( #~a # #~b ) # #~c _ = _ λ #~a + &mu. #~b , _ some λ, &mu. &in. &reals.

Now _ (( #~a # #~b ) # #~c ) &dot. #~c _ = _ [ ( #~a # #~b ), #~c , #~c ] _ = _ 0 , _ so _ λ (#~a &dot. #~c) + &mu. (#~b &dot. #~c) _ = _ 0 _ => _ λ _ = _ - &mu. (#~b &dot. #~c) ./ (#~a &dot. #~c)
i.e. _ ( #~a # #~b ) # #~c _ = _ &mu. ( #~b - ( (#~b &dot. #~c) ./ (#~a &dot. #~c) ) #~a )

Put _ ( #~a # #~b ) # #~c _ = _ ~K_{#~c} ( (#~a &dot. #~c) ~#b - (#~b &dot. #~c) #~a ) _ _ _ where _ ~K_{#~c} _ = _ &mu. ./ (#~a &dot. #~c) _ _ _ _ ... (#1)

In particular _ ( #~a # #~b ) # #~a _ = _ ~K_{#~a} ( (#~a &dot. #~a) ~#b - (#~b &dot. #~a) #~a )
So _ ( #~a # #~b ) # #~a &dot. #~b _ = _ ~K_{#~a} ( (#~a &dot. #~a) ~#b &dot. #~b - (#~b &dot. #~a) #~a &dot. #~b )
_ _ _ ( #~a # #~b ) &dot. ( #~a # #~b ) _ = _ ~K_{#~a} ( #~a ^2 #~b ^2 - (#~a &dot. #~b) ^2 )
_ _ _ #~a ^2 #~b ^2 sin ^2 &theta. _ = _ ~K_{#~a} #~a ^2 #~b ^2 ( 1 - cos ^2 &theta. )
_ _ => _ ~K_{#~a} = 1 _ => _ ( #~a # #~b ) # #~a _ = _ (#~a &dot. #~a) ~#b - (#~b &dot. #~a) #~a _ _ _ _ ... (#2)

From (#1) ... _ ( #~a # #~b ) # #~c &dot. #~a _ = _ ~K_{#~c} ( (#~a &dot. #~c) ~#b &dot. #~a - (#~b &dot. #~c) #~a &dot. #~a )
but _ ( #~a # #~b ) # #~c &dot. #~a _ = _ [ ( #~a # #~b ), #~c, #~a ] _ = _ - [ ( #~a # #~b ), #~a, #~c ] _ = _ - ( #~a # #~b ) # #~a &dot. #~c _ = _ - (#~a &dot. #~a) ~#b &dot. #~c + (#~b &dot. #~a) #~a &dot. #~c _ _ (from (#2)) => _ ~K_{#~c} = 1 _ for any #~c, _ which gives us the result

( #~a # #~b ) # #~c _ = _ (#~a &dot. #~c) ~#b - (#~b &dot. #~c) #~a

Note that _ #~a # ( #~b # #~c ) _ = _ - ( #~b # #~c ) # #~a _ = _ (~#a &dot. #~c) #~b - (~#a &dot. #~b) #~c