Vector (or Cross) Product of two vectors
The vector product
of two vectors 
and 
,
denoted by 
, is defined as the vector 
, where q is the angle between the vectors and and
is a unit vector perpendicular to
both and (i.e.,
perpendicular to the plane of and ).The sense ofis
obtained by the right hand thumb rule i.e., 
and form a
right-handed screw. If we curl the fingers of our right hand from to through
the smaller angle (keeping the initial point of and
same), the thumb points in the
direction of . In this case, ,and
(or ),
in that order are said to form a right handed system.
It is evident that 
=
absinq
Properties:
r 
&⇒ 
r 
(non-commutative)
r 
(Distributive)
r 
r 
Û 
are
collinear (if none of 
is a zero vector)
r 
r 
r If 
then 
= 
r Any vector
perpendicular to the plane of 
is l ()
where l is a real number.
Unit vector perpendicular to is ± 
r denotes the area of
the parallelogram OACB, whereas
area
of DOAB = 
Area is also treated as
a vector with its direction in the proper sense. .
Example.1
If 
and 
show that 
is
parallel to 
.
Solution
We are given that 
and
&⇒ 
&⇒
&⇒ 
are
parallel.
Example.2
If are
vectors from the origin to three points A, B, C show that
is perpendicular to
the plane ABC..
Solution
are the p.v. of the points A, B, C
respectively, Vector 
and 
.
The vector perpendicular to the plane of 
is
along 
= 
= 
.
Example.3
Solve for the
equation provided that is not perpendicular to ..
Solution
We are given that 
&⇒
Hence 
are parallel
&⇒ 
&⇒ 
&⇒ t = 
Hence 
Example.4
Show that the area
of the triangle formed by joining the extremities of an oblique side of a
trapezium to the mid-point of opposite side is half that of the trapezium..
Solution
Let ABCD be the
trapezium and E be the mid-point of BC. Let A be the initial point and let be the p.v. of B and 
that of D. Since DC is parallel to AB,
is a vector along DC, so that the p.v.
of C is 
.
&⇒ the p.v. of E is 
Area of D AED = 
Area of the
trapezium = Area (DACD) + Area (DABC)
= 
= 
= 2DAED.
