27
© NCERT
not to be republished
 MATHEMATICS
452
Let 
aand
rb
r
 be two vectors given in component form as 
1
2
3 ˆ
ˆ
ˆ
a i
a j
a k
+
+
and
1
2
3 ˆ
ˆ
ˆ
b i
b j
b k
+
+
, respectively Then their cross product may be given by
a
b
×
r
r
 =
1
2
3
1
2
3
ˆ
ˆ
ˆ
i
j
k
a
a
a
b
b
b
Explanation We have
a b
×
r
r
 =
1
2
3
1
2
3
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
(
)
(
)
a i
a j
a k
b i
b j
b k
+
+
×
+
+
=
1 1
1 2
1 3
2 1
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
(
)
(
)
(
)
(
)
a b i
i
a b i
j
a b i
k
a b
j
i
×
+
×
+
×
+
×
+ 
2 2
2 3
ˆ
ˆ
ˆ
ˆ
(
)
(
)
a b
j
j
a b
j
k
×
+
×
+  
3 1
3 2
3 3
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
(
)
(
)
(
)
a b k
i
a b k
j
a b k
k
×
+
×
+
×
(by Property 1)
=
1 2
1 3
2 1
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
(
)
(
)
(
)
a b i
j
a b k
i
a b i
j
×
−
×
−
×
+  
2 3
3 1
3 2
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
(
)
(
)
(
)
a b
j
k
a b k
i
a b
j
k
×
+
×
−
×
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
(as
0  and  
,
  and  
)
i
i
j
j
k
k
i
k
k
i
j
i
i
j
k
j
j
k
× =
×
=
×
=
×
= − ×
× = − ×
×
= − ×
=
1 2
1 3
2 1
2 3
3 1
3 2
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
a b k
a b j
a b k
a b i
a b j
a b i
−
−
+
+
−
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
(as
,
  and  
)
i
j
k
j
k
i
k
i
j
×
=
×
=
× =
=
2 3
3 2
1 3
3 1
1 2
2 1 ˆ
ˆ
ˆ
(
)
(
)
(
)
a b
a b i
a b
a b
j
a b
a b k
−
−
−
+
−
=
1
2
3
1
2
3
ˆ
ˆ
ˆ
i
j
k
a
a
a
b
b
b
Example 22 Find 
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
|
|,  if  
2
3   and  
3
5
2
a b
a
i
j
k
b
i
j
k
×
=
+
+
=
+
−
r
r
r
r
Solution We have
a
b
×
r
r
 =
ˆ
ˆ
ˆ
2
1
3
3
5
2
i
j
k
−
=
ˆ
ˆ
ˆ
( 2
15)
( 4
9)
(10 – 3)
i
j
k
− −
− − −
+
ˆ
ˆ
ˆ
17
13
7
i
j
k
= −
+
+
Hence
|
|
 
r
ra b  =
2
2
2
( 17)
(13)
(7)
507
−
+
+
=
© NCERT
not to be republished
VECTOR ALGEBRA
453
Example 23 Find a unit vector perpendicular to each of the vectors (
)
a
b
+
r
r
 and
(
),
a
−b
r
r
where 
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
,
2
3
a
i
j
k
b
i
j
k
= +
+
= +
+
r
r Solution We have 
ˆ
ˆ
ˆ
ˆ
ˆ
2
3
4
and
2
a
b
i
j
k
a
b
j
k
+
=
+
+
−
= − −
r
r
r
r
A vector which is perpendicular to both 
and
a
b
a
b
+
−
r
r
r
r
 is given by
(
)
(
)
a
b
a
b
+
×
−
r
r
r
r
 =
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
2
3
4
2
4
2
(
, say)
0
1
2
i
j
k
i
j
k
c
= −
+
−
=
−
−
r
Now
|
|
cr  =
4
16
4
24
2 6
+
+
=
=
Therefore, the required unit vector is
|
c|
c
r
r  =
1
2
1 ˆ
ˆ
ˆ
6
6
6
i
j
k
−
+
−
�Note   There are two perpendicular directions to any plane Thus, another unit
vector perpendicular to 
and
a
b
a
b
+
−
r
r
r
r
 will be 1
2
1 ˆ
ˆ
ˆ