MATH1110 Lecture Notes - Lecture 9: Unit Vector, Cross Product, Scalar Multiplication
18 Jun 2018
School
Department
Course
Professor
MATH1110: Mathematics 1
Vectors (III): The Cross Product and Triple
Products
1 Definition of the Cross Product
We have already seen the vector operations of vector addition, scalar multi-
plication and the dot product. Another operation on vectors that has proved
useful in a large number of contexts is called the cross (or vector) product.
Geometric definition: The cross product of two 3-dimensional vectors
uand v, denoted by u×v, is a vector of length
kukkvksin(θ)
and direction
b
n
for θand b
ndefined as follows.
•θis the angle between uand vthat satisfies 0 ≤θ≤π.
•b
nis the unit vector which is perpendicular to both uand v, which
satisfies the right-hand-rule.
The right-hand-rule, illustrated overpage, says that if you orient your
right hand such that your fingers point straight out in the direction of
vector uand your palm is oriented to rotate towards vector v, then your
thumb will be pointing in the direction of the cross product.
Note that we will soon see an algebraic definition of the cross product
which does not require the right-hand-rule to define it, however the geometric
definition, whilst difficult to state, is very intuitive for applications, especially
those related to the Physics of magnetic fields and electric currents.
Page 1
Document Summary
We have already seen the vector operations of vector addition, scalar multi- plication and the dot product. Another operation on vectors that has proved useful in a large number of contexts is called the cross (or vector) product. The right hand rule illustrated: u x v u v. Note the the right hand rule is used to choose between two di erent possible choices for the unit vector which is perpendicular to both u and v. The set of vectors {u, v, n} in gure (a) are said to form a right-handed system whereas the set in gure (b) are said to form a left-handed system. (a) (b) Convince yourself that the right-hand-rule gives you a choice compatible with gure (a) and not with gure (b). For the vectors i, j and k calculate i j. The length of i j is i j = ||i|| ||j|| sin(cid:16) .
