vector

Vectors 375
5.11 VECTOR PRODUCT OR CROSS PRODUCT
1.
The vector, or cross product of two vectors a
and b is defined to be a vector such that
(i) Its magnitude is
a
b
sin , where is the
b
angle between a and b .
(ii) Its direction is perpendicular to both vectors
a and b
.
(iii) It forms with a right handed system.
Let ^
be a unit vector perpendicular to both the vectors a and b .
a b =
2. Useful results
a
b
sin .
Since ^i , ^j , ^k are three mutually perpendicular unit vectors, then
^
^
^ ^ ^ ^
i i = j j k k 0
^
^
i j =
^ ^ ^
j i k
^ ^ ^ ^
j i i j
^ ^ ^
ĵ k ˆ = k j
^ ^ ^ ^
i and k j j k
^ ^ ^
kˆ
i ˆ
^ ^ ^ ^
= i k j i k k i
5.12 VECTOR PRODUCT EXPRESSED AS A DETERMINANT
If
^ ^ ^
a = a i a j a k
1 2 3
^ ^ ^
b = b1 i b2 j b3
k
a b =
=
^ ^ ^ ^ ^ ^
1 2 3 1 2 3
( a i a j a k) ( b i b j b k)
^ ^ ^ ^ ^ ^ ^ ^ ^ ^
1 1( ) 1 2( ) 1 3( ) 2 1( ) 2 2 ( )
^ ^ ^ ^ ^ ^ ^ ^
ab 2 3 j k ab 3 1k i ab 3 2 k j ab 3 3 k k
ab i i ab i j ab i k ab j i ab j j
^ ^ ^ ^ ^ ^
1 2 1 3 2 1 2 3 3 1 3 2
= ab k a b j ab k a b i ab j a b i
=
=
^ ^ ^
2 3 3 2 1 3 3 1 1 2 2 1
^ ^ ^
( ab ab) i ( a b a b ) j ( ab ab)
k
i j k
a a a
1 2 3
b b b
1 2 3
5.13 AREA OF PARALLELOGRAM
( ) ( ) ( ) ( )
Example 3. Find the area of a parallelogram whose adjacent sides are i – 2j + 3 k and
2i + j – 4k.
^ ^ ^
i j k
Solution. Vector area of gm = 1 2 3
2 1 4
a