Assigned work:
pg.407 #1-13
Recall dot product produces a scalar from
two vectors.
Today we examine a Cross Product (or
Vector Product) which produces a vector
from two vectors.
7.6 Cross Product
Cross Product of a and b will be a vector that is
perpendicular to both a and b . Therefore the cross
product is ONLY defined in R3.
It is useful in physical problems such as torque and
area of a parallelogram (applications we will discuss
tomorrow)
7.6 Cross Product
Prove Cross Product formula :
7.6 Cross Product
Cross Product of a  (a1 , a2 , a3 ) and b  (b1, b2 , b3 )
is the vector:
a  b  (a2b3  a3b2 , a3b1  a1b3 , a1b2  a2b1 )
7.6 Cross Product
An easier method to remember the Cross
Product formula is:
  a2
ab  
  b2
where
a3   a3
,
b3   b3
a b

  ad  bc
c d
a1   a1
,
b1   b1
a2  

b2  
7.6 Cross Product
Finding a Vector Perpendicular to Two Vectors:
If a and b are two non-collinear vectors in 3D,
then every vector perpendicular to both a and b
is of the form k (a  b) where k  R
7.6 Cross Product
Ex 1:
a) Find a vector perpendicular to the vectors
(2,5,0) and (-4,0,9).
Answer: (45,-18,20)
b) Check your answer using Dot Product (since
dot product of 2 perpendicular vectors should
be 0).
7.6 Cross Product
Magnitude of the Cross Product of a and b is:
a  b  a b sin 
where is the angle between a and b
and 0    180
7.6 Cross Product
Direction of Cross Product of a  b is such that:
a, b, and a  b form what we call a right handed
system……
Place your right hand on the diagram so that your
finger curled in the direction from a to b is an angle
less than 180 . The direction of a  b will be the
direction your thumb points.
7.6 Cross Product
Direction of Cross Product of a  b :
a
Fingers curl
this way
b
Thumb is in so:
a  b is into page
b
Fingers curl
this way
a
Thumb is out so:
a  b is out of page
7.6 Cross Product
Direction of Cross Product of a  b :
Note there are other methods of this right hand
rule to find the direction. Some of you may be
used to using the first vector as the thumb, the
second vector as your fingers and the direction of
the cross product as the palm of your hand.
(Motor right hand rule in Grade 11 Physics)
Either method works – use which one you like the
best.
7.6 Cross Product
Ex 2:
If a  5 and b  6 and the angle between a and b
is 45 degrees Determine the magnitude and
direction of a  b . (include a diagram).
Answer:
Magnitude is 15 2 and direction will depend on
how you drew your diagram.
7.6 Cross Product
Ex 3:
Find the cross product of :
a  i  3 j  2k and b  4i  6 j  7k
Answer:
33i  1 j 18k
7.6 Cross Product
Properties of Cross Product:
Let a, b, and c be vectors in 3D


1) a  b   b  a Order matters (anti-commutative)

 

  
 

2) a  b  c  a  b  a  c Distributive Law
 
3) k a  b  ka  b  a  kb
7.6 Cross Product
Ex 4:
Given: a  (6, 2, 3) b  (5,1, 4) c  (4, 2,1)
Determine:
a  b  c
(Note: Think about what order it should be done. Cross
product MUST go 1st since you cannot do cross product of
a vector and a scalar)
Answer: 78
7.6 Cross Product
Important READ 7.6 before doing the assigned
questions.
Make a note on:
1) What is a “Triple Scalar Product”?
2) What is a Triple Vector Product”?