vector

388 Vectors
i j k
=
x y z
x y z
are represented in the adjoining figure.
^
i
^
k
^
m . i
k
r
j
O
O
Number line
j 0 1 2 3 O
i
i
(i) (ii) (iii) (iv)
m r
Example 16. If = 3x 2 y – y 3 z 2 ; find grad at the point (1, –2, –1).
(AMIETE, June 2009, U.P., I Semester, Dec. 2006)
Solution. grad =
2 3 2
= i j k (3 x y y z )
x y z
2 3 2
2 3 2 2 3 2
= i (3 x y y z ) j (3 x y yz) k (3 x y y z )
x y z
=
= 0
2 2 2 3
i (6 xy) j (3x 3 y z ) k ( 2 yz)
grad at (1, –2, –1) = i (6) (1) ( 2) j[(3) (1) 3(4) (1)] k ( 2)( 8) ( 1)
= 12 i 9 j 16 k
Ans.
Example 17. If u = x + y + z, v = x 2 + y 2 + z 2 , w = yz + zx + xy prove that grad u,
grad v and grad w are coplanar vectors. [U.P., I Semester, 2001]
Solution. We have,
grad u =
i j k ( x y z)
i j k
x y z
grad v =
2 2 2
i j k ( x y z ) 2xi 2yj
2zk
x y z
grad w =
i j k ( yz zx xy) i( z y) jz ( x) k( y x)
x y z
[For vectors to be coplanar, their scalar triple product is 0]
Now, grad u.(grad v × grad w) =
=
=
1 1 1 1 1 1
2x 2y 2z 2 x y z
z y z x y x z y z x y x
1 1 1
2 x y z x y z x y z
z y z x y x
1 1 1
2( x y z) 1 1 1 0
y z z x x y
k
[Applying R 2
R 2
+ R 3
]
j