vector

Vectors 385
Normal to (2), 2
= (z – x 2 – y 2 + 3)
=
ˆ
x y z
ˆ ˆ 2 2
i j k ( z– x – y 3)
= – 2 x î – 2 y ĵ + ˆk
Normal to (2) at (2, – 1, 2), 2
= – 4î + 2 ĵ + ˆk ...(4)
1.
2 = | 1|| 2|cos
1.
2
(4 iˆ –2ˆj 4 kˆ).(– 4iˆ 2 ˆj kˆ) –16 –44
cos = =
=
| 1|| 2|
|4 iˆ –2ˆj 4 kˆ||– 4iˆ 2 ˆj kˆ|
16 416 16 41
= –16 –8
=
6 21 3 21
–1 –8
= cos
3 21
–1 –8
Hence the angle between (1) and (2) cos
Ans
3 21
EXERCISE 5.6
2
3
t
t
1. The coordinates of a moving particle are given by x = 4t and y = 3 6 t . Find the
2
6
velocity and acceleration of the particle when t = 2 secs. Ans. 4.47, 2.24
2. A particle moves along the curve
x = 2t 2 , y = t 2 – 4t and z = 3t – 5
where t is the time. Find the components of its velocity and acceleration at time t = 1, in the
8 14 14
direction i 3 j 2 k.
(Nagpur, Summer 2001) Ans. ,
7 7
3. Find the unit tangent and unit normal vector at t = 2 on the curve x = t 2 – 1, y = 4t – 3,
z = 2t 2 – 6t where t is any variable.
Ans.
1 1
(2 i 2 j k), (2i 2 k)
3 3 5
d
dG d F
4. Prove that ( F G)
F G
dt dt dt
5. Find the angle between the tangents to the curve 2 3
r t i 2 t j t k,
at the points t = ± 1.
Ans. 1 9
cos
17
6. If the surface 5x 2 – 2byz = 9x be orthogonal to the surface 4x 2 y + z 3 = 4 at the point (1, –1, 2)
then b is equal to
(a) 0 (b) 1 (c) 2 (d) 3 (AMIETE, Dec. 2009) Ans. (b)
5.25 SCALAR AND VECTOR POINT FUNCTIONS
Point function. A variable quantity whose value at any point
N
^
in a region of space depends upon the position of the point, is
R
Q
called a point function. There are two types of point functions.
n r
(i) Scalar point function. If to each point P (x, y, z) of a
region R in space there corresponds a unique scalar f (P), then f is
P
called a scalar point function. For example, the temperature
distribution in a heated body, density of a body and potential due to gravity are the examples of
a scalar point function.
(ii) Vector point function. If to each point P (x, y, z) of a region R in space there corresponds
a unique vector f (P), then f is called a vector point function. The velocity of a moving fluid,
gravitational force are the examples of vector point function.
(U.P., I Semester, Winter 2000)
= c
+ d = c