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Basic Concepts Of Vector Part 5

Posted on - 04-01-2017

Math

IIT JEE

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.

 
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