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
in that order are said to form a right handed system.
It is evident that =
r Û are
collinear (if none of is a zero vector)
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
of DOAB =
Area is also treated as
a vector with its direction in the proper sense. .
If and show that is
parallel to .
We are given that and
vectors from the origin to three points A, B, C show that
is perpendicular to
the plane ABC..
are the p.v. of the points A, B, C
respectively, Vector and .
The vector perpendicular to the plane of is
Solve for the
equation provided that is not perpendicular to ..
We are given that
Hence are parallel
&⇒ t =
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..
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.