Vectors - Part 2: Products
Dot Product (scalar product)
r r
If the angle between two vectors A and B is θ, then we define the dot product
r r
A ⋅ B = AB cos(θ )
This product is a scalar. The angle between two vectors is always between 0
and 180°.
This product obeys the commutative and distributive rules.
Special cases:
r r
θ = 0 (parallel vectors): A ⋅ B = AB (the product of the magnitudes, positive)
r r
θ = 90° (perpendicular): A⋅ B = 0
r r
θ = 180° (anti-parallel): A ⋅ B = − AB (negative)
r r
A ⋅ A = A2 (any vector, dotted with itself, = the magnitude squared)
Applied to unit vectors:
iˆ ⋅ iˆ = ˆj ⋅ ˆj = kˆ ⋅ kˆ = 1 iˆ ⋅ ˆj = ˆj ⋅ kˆ = kˆ ⋅ iˆ = 0
This makes it easy to find the dot product of vectors in component form. For
example, ( )( )
2iˆ − 3 ˆj ⋅ 5iˆ + ˆj = 2 ⋅ 5 − 3 ⋅ 1 = 7
One use of the dot product is to find the angle between two vectors. From the
definition above, the angle θ between two vectors is found from
r r
A⋅ B
cos(θ ) =
AB
As an example, let’s find the angle between the face diagonal and body diagonal
of a cube. Let the cube have side = s with the edges aligned in a Cartesian CS.
Then we can represent the face diagonal as siˆ + sjˆ and the body diagonal as
r r r r
siˆ + sjˆ + skˆ . If these vectors are respectively A and B , then A ⋅ B = 2 s 2 ,
2
(magnitudes) A = 2 s , and B = 3s . Thus, cos(θ ) = , and θ = 35.3°.
6
Another use of the dot product is that it gives us a new way to express the
r r r r
magnitude of a vector. Since A ⋅ A = A2 , then A = A ⋅ A .
The dot product is useful in proving the law of cosines – namely that if θ is the
exterior angle between two sides of a triangle, A and B, and C is the 3rd side,
then C2 = A2 + B2 + 2 A b cos(θ). To see this, consider a triangle of
r r r
vectors expressing the addition A + B = C . Now, square both sides (meaning,
since the sides are vectors, to dot each side with itself)
r r r r r r
( A + B) ⋅ ( A + B) = A
2
+ B 2 + 2 A ⋅ B = A2 + B 2 + 2 AB cos(θ ) = C 2
Cross Product (vector product)
This is a different way to multiply vectors, yielding a product which is a vector. If
r r r
the angle between two vectors A and B is θ, then we define C , the cross
r r r r r
product of A and B , C = A × B as a vector of magnitude C = A B sin(θ),
r r
whose direction is perpendicular to the plane defined by A and B . This does
not completely define the direction, since there are two (opposite) such
directions. In order to choose, we say in the right-hand sense. This means that
we use a right-hand rule to specify which of the two possible directions we mean.
Properties:
Cross products are necessarily three-dimensional. Because of the RH rule, the
r r r r
multiplication is not commutative A × B = − B × A (it is anticommutative), and not
r r r r r r
( ) ( )
associative A × B × C ≠ A × B × C , although it does satisfy the distributive law.
Special Cases:
r r
θ = 0 or 180° : A × B = 0 (sin(θ) = 0)
r r
θ = 90° (perpendicular): A × B = AB
r r
A× A = 0
Unit Vectors:
iˆ × iˆ = ˆj × ˆj = kˆ × kˆ = 0 iˆ × ˆj = kˆ ˆj × kˆ = iˆ kˆ × iˆ = ˆj this assumes a RH
co-ordinate system
As with the dot product, the unit vector products make it easy to cross two
vectors in component form. For example,
( ) ( ) ( ) ( ) ()
2iˆ − 3 ˆj × iˆ + 5kˆ = 2 ⋅ 5 ⋅ − ˆj − 3 ⋅ 1 ⋅ − kˆ − 3 ⋅ 5 ⋅ iˆ = −15iˆ − 10 ˆj + 3kˆ
The cross product can be used to prove the law of sines: in the triangle,
B
γ
α
A
C
β
sin(α ) sin( β ) sin(γ )
= =
A B C
r r r
By making the sides into the vector sum A + B + C = 0 , we can cross one vector
r r r r r r r r
( )
with the sum: A × A + B + C = A × B + A × C . Taking magnitudes,
AB sin(γ ) − AC sin( β ) = 0 , divide by ABC to get the law of sines.
