Test: Vector Analysis

QUESTION: 1

Divergence and Curl of a vector field are _________.

A.
Scalar & Scalar
B.
Scalar & Vector
C.
Vector & Scalar
D.
Vector & Vector

Solution:

QUESTION: 2

The two vector are R_AM = 20u_x + 18u_y - 10u_zand R_AN = -10u_x + 8u_y + 15u_z. The u of the triangle that bisects the interior angle at A is:

A.
0.168u_x + 0.915u_y + 0.367u_z
B.
0.729u_x + 0.134u_y - 0.672u_z
C.
0.729u_x + 0.134u_y + 0.672u_z
D.
0.168u_x + 0.915u_y - 0.367u_z

Solution:

The non-unit vector in the required direction is:

QUESTION: 3

Two points in cylindrical coordinates are A(ρ = 5, Ø = 70⁰, z = -3) and B(ρ = 2, Ø = 30⁰, z = 1) A unit vector at A towards B is:

A.
0.03u_x - 0.82u_y + 0.57u_z
B.
0.03u_x + 0.82u_y + 0.57u_z
C.
- 0.03u_x + 0.82u_y + 0.57u_z
D.
0.003u_x - 0.82u_y + 0.57u_z

Solution:

In cartesian coordinates:

QUESTION: 4

The surface ρ = 2, ρ = 4, Ø = 100⁰ ,Ø = 130⁰ ,z = 3 and z = 4.5 define a closed surface. The enclosed volume is:

A.
480
B.
5.46
C.
360
D.
6.28

Solution:

QUESTION: 5

The surface ρ = 2, ρ = 4, Ø = 45⁰ ,Ø = 135⁰ , z = 3 and z = 4 define a closed surface. The total area of the enclosing surface is:

A.
34.29
B.
20.7
C.
32.27
D.
16.4

Solution:

Area is:

QUESTION: 6

The surface ρ = 3, ρ = 5, Ø = 100⁰ ,Ø = 130⁰ , z = 3 and z = 4.5 define a closed surface. The length of the longest straight line that lies entirely within the volume is:

A.
3.21
B.
3.13
C.
4.26
D.
4.21

Solution:

QUESTION: 7

A vector field H is:

A.
0.25
B.
0.433
C.
-0.433
D.
-0.25

Solution:

QUESTION: 8

A vector is A = yu_x +(x+z)u_y .At point P(-2, 6, 3) A in cylindrical coordinate is:

A.
-0.949u_ρ - 6.008u_Ø
B.
0.949u_ρ - 6.008u_Ø
C.
-6.008u_ρ - 0.949u_Ø
D.
6.008u_ρ + 0.949u_Ø

Solution:

QUESTION: 9

The directional derivative of function φ = xy + yz + zx at point P(3, -3, -3) in the directiontoward point Q(4, -1, -1) is:

A.
-3
B.
1
C.
-2
D.
0

Solution:

QUESTION: 10

The temperature in a auditorium is given by T = 2x² +y² - 2z² . A mosquito located at (2, 2, 1) in the auditorium desires to fly in such a direction that it will get warm as soon as possible. The direction, in that it must fly is: