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Basic Facts - Dot Products and Cross Products In addition to the most basic operations of scaling and vector addition (both done component-wise), the measurement of lengths and angles are facilitated by the dot product of vectors (also known as the inner product). The dot product can be defined in R n for any n which allows for the definition of orthogonality in any
dimension. In addition, the cross product (defined only in R^3 ) allows us construct a vector orthogonal to any pair of vectors and also to measure areas of parallelograms.
Definition : The dot product of two vectors u , v in R n is a scalar defined as follows:
u v u 1 (^) , u 2 (^) , , un v 1 (^) , v 2 (^) , , vn u v 1 1 (^) u v 2 2 u vn n
There are some easy-to-verify algebraic properties of the dot product that follow from this definition:
Algebraic Properties of the Dot Product : Suppose u , v , and w are vectors in R n and that t is any scalar.
- v u u v (symmetry, dot product is commutative)
^ ^ ^ ^ ^
^ ^ ^ ^ ^
u v w u v u w u v w u w v w
(left and right distributive laws)
( t u ) v t ( u v ) u ( t v ) (how the dot product behaves relative to scaling of vectors)
2 2 u u u 0 for all u (and u u u 0 only for u 0 )
Using these algebraic properties and the Law of Cosines (a corollary of the Pythagorean Theorem) we were able to derive the following important property of the dot product:
If u , v in R n are two vectors emanating out from a common vertex to form an angle , and if u and v are
their respective lengths, then u v u v cos . The great importance of this relation is that connects the
algebraically-defined dot product to the geometric measurements of lengths and angles.
We immediately get the following corollary using some basic trigonometric facts: If u , v in R n are nonzero
vectors emanating from a common vertex to form an angle , then
u v 0 if and only if the angle is acute
u v 0 if and only if the angle is obtuse
u v 0 if and only if the angle is a right angle, i.e. u v
We can also use the relation u v u v cos (and a sketch) to define the scalar projection of a vector v
in the direction of another vector u (also called the component of v in the direction of u ) as
u v u
. This is
perhaps best remembered by noting that to find the component of a vector v in any given direction, you “dot v with a unit vector in that direction”. We can then use this fact to define the vector projection of v in the
direction of u by construction it as Proj (^2)
^
^
u
u u v u v v u u u (^) u
. This can be useful for expressing a vector
as the sum of a “tangential component” vector and a “normal component” vector, especially in geometry and physics.
The fact that the orthogonality of vectors can be characterized algebraically by their dot product being zero allowed us to derive that the equation of a plane with normal vector n and passing through a point with position
vector x 0 must be of the form n ( x x (^) 0 ) 0 where x represents the position vector of any other point on the
plane. In R^3 , if we express this in components with n A B C , , , x (^) 0 x 0 (^) , y 0 (^) , z 0 , and x x , y z , , this
becomes A x ( x 0 (^) ) B y ( y 0 (^) ) C z ( z 0 ) 0 or Ax By Cz D where D is the constant obtained after
multiply out and transposing constants to the right-hand-side. In problems, we often jump to this form once we know the normal vector and determine D by plugging in the coordinates of the given point.
It is sometimes the case that we need to find the equation of a plane given not a normal vector and a single point, but rather three non-colinear points. In this case, we can take points pairwise to produce vectors parallel to the plane and may desire to use these to find a vector orthogonal to the plane. A convenient way to do this is
via the cross product (defined only in R^3 ). Given two vectors u u 1 (^) , u 2 (^) , u 3 and v v 1 (^) , v 2 (^) , v 3 in R^3 , we can
use the orthogonality requirement to show that the following cross product will be orthogonal to both vectors:
u v u 1 (^) , u 2 (^) , u 3 (^) v 1 (^) , v 2 (^) , v 3 (^) u v 2 3 (^) u v 3 2 (^) , u v 3 1 (^) u v 1 3 (^) , u v 1 2 (^) u v 2 1
There are several different ways to express this using the definition of a 2 2 determinant, namely
det a b a b ad bc c d c d
. Examining the above expression we see that:
2 3 3 1 1 2 2 3 1 3 1 2 2 3 3 1 1 2 2 3 1 3 1 2
u u u u u u u u u u u u v v v v v v v v v v v v u v
Note the sign switch in the middle component. This is done so that you can conveniently perform the
calculation by creating a 2 3 array from the given two vectors 1 2 3 1 2 3
u u u v v v
and then respectively covering the
1st, 2nd, and 3rd columns and calculating the determinant of the resulting 2 2 determinants (with appropriate
sign switch of the middle component. For example, if u 1,3, 6 and v 2,5, 4 , we would get the array
1 3 6 2 5 4
(^) and use the procedure to calculate^ u^ ^ v^ ^12 ^ 30,^ (4^ 12),5^ ^6 ^ 18,^ 16,11. A quick check
using the dot product shows that this is orthogonal to both u and v.
Some people prefer to express this procedure using i j k , , notation by formally calculating the 3 3
determinant 1 2 3 2 3 1 3 1 2 (^2 3 1 3 1 ) 1 2 3
u u u u u u u u u v v v v v v v v v
i j k i j k.
Using only this algebraic definition for the cross product, we can derive the following properties:
Algebraic Properties of the Cross Product : Suppose u , v , and w are vectors in R^3 and that t is any scalar.
- v u u v (anticommutative) [Corollary: u u 0 for any vector u ]
^ ^ ^ ^ ^
^ ^ ^ ^ ^
u v w u v u w u v w u w v w
(left and right distributive laws)
( t u ) v t ( u v ) u ( t v ) (how the dot product behaves relative to scaling of vectors)
u 0 0
u ( v w ) ( u v ) w (triple scalar product)
u ( v w ) ( u w v ) ( u v w ) (triple vector product)
