vector

Vectors 393
Velocity, v =
dr
2
3t i 2tj
dt
Velocity at t = 1 is = 3i
2 j
The component of velocity in the same direction of velocity
3i
2 j
9
4
=
(3 i 2 j). 13
9
4 13
Ans.
Example 26. Find the directional derivative of (x, y, z) = x 2 y z + 4 x z 2 at (1, –2, 1) in
the direction of 2i j 2 k . Find the greatest rate of increase of .
(Uttarakhand, I Semester, Dec. 2006)
Solution. Here, (x, y, z) = x 2 y z + 4xz 2
Now, =
=
at (1, – 2, 1) =
2 2
i j k ( x yz 4 xz )
x y z
2 2 2
(2xyz 4 z ) i ( x z) j ( x y 8 xz)
k
2
{2(1) ( 2)(1) 4(1) } i (11) j {1( 2) 8(1)(1)} k
= ( 4 4) i j ( 2 8) k = j 6 k
Let a = unit vector = 2 i j 2 k 1
(2 i j 2 k)
41
4 3
So, the required directional derivative at (1, –2, 1)
1
= . a ( j 6 k). (2 i j 2 k)
= 1
( 112)
13
3
3 3
Greatest rate of increase of = j 6k
= 1
36
= 37 Ans.
Example 27. Find the directional derivative of the function = x 2 – y 2 + 2z 2 at the point P
(1, 2, 3) in the direction of the line PQ where Q is the point (5, 0, 4).
(AMIETE, Dec. 20010, Nagpur University, Summer 2008, U.P., I Sem., Winter 2000)
Solution. Directional derivative =
=
i j k ( x y 2 z ) 2xi 2yj
4zk
x y z
2 2 2
Directional Derivative at the point P (1, 2, 3) = 2i 4 j 12 k
...(1)
PQ = Q P = (5, 0, 4) – (1, 2, 3) = (4, –2, 1) ...(2)
(4 i 2 j k)
Directional Derivative along PQ = (2 i 4 j 12 k).
[From (1) and (2)]
16 41
=
8812 28
21 21
Ans.
x
Example 28. For the function (x, y) = 2 2 , find the magnitude of the directional
x y
derivative along a line making an angle 30° with the positive x-axis at (0, 2).
(A.M.I.E.T.E., Winter 2002)