A Vector in The Direction of Vector That Has Magnitude 15 Is

Chapter-1
Vector Analysis
Type-1 Questions:
1. A vector in the direction of vector i−2

^ ^j+2 k^ that has magnitude 15 is
Ans: (d)
2.
(a) 30°
(b) 45°
(c) 60°
(d) 90°
Ans: (a)
3.
Ans (c)
4.
Ans: (b)
5.
6. Write the direction cosines of the vector −2 i+

^ ^j−5 k^
7. The value of X for which vectors are orthogonal is

__________ .
Ans:
8. Transform the vector B=yi+(x+z)j located at point (-2,6,3) into cylindrical

coordinates.
a) (6.325,-71.57,3)
b) (6.325,71.57,3)
c) (6.325,73.57,3)
d) (6.325,-73.57,3)
Answer: a
Explanation: ρ = √(x2+y2) = √40 = 6.325
Φ = tan-1(y/x) = tan-1(-6/2) = -71.57
z = 3.
9. Convert the given rectangular coordinates A(2,3,1) into corresponding cylindrical
coordinates
a) (3.21,56.31,1)
b) (3.21,57.31,0)
c) (3.61,57.31,0)
d) (3.61,56.31,1)
Answer: d
Explanation: ρ = √(x2+y2) = √13 = 3.61
Φ = tan-1(y/x) = 56.31
z=1
Thus, A = (3.61,56.31,1)
10. Transform the spherical system B = (10sin θ/r + rcos2θ)i + j + (10cos θ/r –r sin θ cos
θ)k into cylindrical form at (5, π/2, -2)
a) 2.467i + j + 1.167k
b) 2.467i – j + 1.167k
c) 2.467i – j – 1.167k
d) 2.467i + j – 1.167k
Answer: a
The equivalent cylindrical form is given by,

B = (10sin θ/r + rcos2θ)i + j + (10cos θ/r –r sin θ cos θ)k
At (5, π/2, -2), r = √(52+-22) = √29
sin θ = 5/√29 and cos θ = -2/√29
Thus, B = 2.467i + j + 1.167k.
11. Convert the point (3,4,5) from Cartesian to spherical coordinates

a) (7.07,45⁰,53⁰)
b) (0.707,45⁰,53⁰)
c) (7.07,54⁰,63⁰)
d) (0.707,54⁰,63⁰)
Answer: a
Explanation: r = √(x2+y2+z2) = √50 = 7.07
Θ = cos-1(z/r) = cos-1(5/5√2) = 45⁰
Φ = tan-1(y/x) = tan-1(4/3) = 53⁰.
12. Find the spherical coordinates of A(2,3,-1)
a) (3.74, 105.5⁰, 56.13⁰)
b) (3.74, 105.5⁰, 56.31⁰)
c) (3.74, 106.5⁰, 56.13⁰)
d) (3.74, 106.5⁰, 56.31⁰)
Answer: b
Explanation: r = √(x2+y2+z2) = √14 = 3.74
Θ = cos-1(z/r) = cos-1(-1/3.74) = 105.5⁰
Φ = tan-1(y/x) = tan-1(3/2) = 56.31⁰
13. Find the Cartesian coordinates of B(4,25⁰,120⁰)
a) (0.845, 1.462, 3.625)

b) (-0.845, 1.462, 3.625)
c) (-8.45, 2.462, 6.325)
d) (8.45, 2.462, 6.325)
Answer: b
Explanation: x = r sin θ cos φ = 4 sin25⁰ cos 120⁰ = -0.845
y = r sin θ sin φ = 4 sin 25⁰ sin 120⁰ = 1.462
z = r cos θ = 4 cos 25⁰ = 3.625.
14. Transform the vector (4,-2,-4) at (1,2,3) into spherical coordinates.

a) 3.197i – 2.393j + 4.472k
b) -3.197i + 2.393j – 4.472k
c) 3.197i + 2.393j + 4.472k
d) -3.197i – 2.393j – 4.472k
Explanation: r = √(x2+y2+z2) = 3.74

Θ = cos-1(z/r) = cos-1(3/3.74) = 36.7⁰
Φ = tan-1(y/x) = tan-1(2/1) = 63.4⁰
A = (4 sin θ cos φ – 2 sin θ sin φ – 4cos θ)i + (4 cos θ cos φ – 2 cos θ sin φ + 4 sin θ)j +
(-4 sin φ – 2 cos φ)k
On substituting r, θ, φ, A = -3.197i + 2.393j – 4.472k.
15. The mathematical perception of the gradient is said to be

a) Tangent
b) Chord
c) Slope
d) Arc
Answer: c
Explanation: The gradient is the rate of change of space of flux in electromagnetics. This
is analogous to the slope in mathematics.
16. Divergence of gradient of a vector function is equivalent to

a) Laplacian operation
b) Curl operation
c) Double gradient operation
d) Null vector
Ans: Answer: a
Explanation: Div (Grad V) = (Del)2V, which is the Laplacian operation. A function is said
to be harmonic in nature, when its Laplacian tends to zero.
17. Find the gradient of t = x2y+ ez at the point p(1,5,-2)

a) i + 10j + 0.135k
b) 10i + j + 0.135k
c) i + 0.135j + 10k
d) 10i + 0.135j + k
Answer: b
Explanation: Grad(t) = 2xy i + x2 j + ez k. On substituting p(1,5,-2), we get 10i + j +
0.135k.
18. Curl of gradient of a vector is

a) Unity
b) Zero
c) Null vector
d) Depends on the constants of the vector
Answer: c
Explanation: Gradient of any function leads to a vector. Similarly curl of that vector gives
another vector, which is always zero for all constants of the vector. A zero value in
vector is always termed as null vector(not simply a zero).
19. The gradient can be replaced by which of the following?

a) Maxwell equation
b) Volume integral
c) Differential equation
d) Surface integral
Answer: c
Explanation: Since gradient is the maximum space rate of change of flux, it can be
replaced by differential equations.
20. When gradient of a function is zero, the function lies parallel to the x-axis. State
True/False.
a) True
b) False
Answer: a
Explanation: Gradient of a function is zero implies slope is zero. When slope is zero, the
function will be parallel to x-axis or y value is constant.
21. The divergence of a vector is a scalar. State True/False.

a) True
b) False
Answer: a
Explanation: Divergence can be computed only for a vector. Since it is the measure of
outward flow of flux from a small closed surface as the volume shrinks to zero, the result
will be directionless (scalar).
22. Determine the divergence of F = 30 i + 2xy j + 5xz 2 k at (1,1,-0.2) and state the nature of
the field.
a) 1, solenoidal
b) 0, solenoidal
c) 1, divergent
d) 0, divergent
Answer: b
Explanation: Div(F) = Dx(30) + Dy(2xy) + Dz(5xz 2) = 0 + 2x + 10xz = 2x + 10xz
Divergence at (1,1,-0.2) will give zero. As the divergence is zero, field is solenoidal.
Alternate/Shortcut: Without calculation, we can easily choose option b, as by

theory when the divergence is zero, the vector is solenoidal. Option b is the only
one which is satisfying this condition.
23. Identify the nature of the field, if the divergence is zero and curl is also zero.
a) Solenoidal, irrotational
b) Divergent, rotational
c) Solenoidal, irrotational
d) Divergent, rotational
Answer: c
Explanation: Since the vector field does not diverge (moves in a straight path), the
divergence is zero. Also, the path does not possess any curls, so the field is irrotational.
24. Curl is defined as the angular velocity at every point of the vector field. State True/False.
a) True
b) False
Answer: a
Explanation: Curl is defined as the circulation of a vector per unit area. It is the cross
product of the del operator and any vector field. Circulation implies the angular at every
point of the vector field. It is obtained by multiplying the component of the vector parallel
to the specified closed path at each point along it, by the differential path length and
summing the results.
25. The curl of curl of a vector is given by,

a) Div(Grad V) – (Del)2V
b) Grad(Div V) – (Del)2V
c) (Del)2V – Div(Grad V)
d) (Del)2V – Grad(Div V)
Answer: b
Explanation: Curl (Curl V) = Grad (Div V) – (Del) 2V is a standard result of the curl
operation.
26. Which of the following theorem use the curl operation?

a) Green’s theorem
b) Gauss Divergence theorem
c) Stoke’s theorem
d) Maxwell equation
Answer: c
Explanation: The Stoke’s theorem is given by ∫ A.dl = ∫Curl(A).ds, which uses the curl
operation. There can be confusion with Maxwell equation also, but it uses curl in
electromagnetics specifically, whereas the Stoke’s theorem uses it in a generalised
manner. Thus the best option is c.

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Chapter-1

1. A vector in the direction of vector i−2

6. Write the direction cosines of the vector −2 i+

7. The value of X for which vectors are orthogonal is

8. Transform the vector B=yi+(x+z)j located at point (-2,6,3) into cylindrical

The equivalent cylindrical form is given by,

11. Convert the point (3,4,5) from Cartesian to spherical coordinates

13. Find the Cartesian coordinates of B(4,25⁰,120⁰)

a) (0.845, 1.462, 3.625)

14. Transform the vector (4,-2,-4) at (1,2,3) into spherical coordinates.

Explanation: r = √(x2+y2+z2) = 3.74

15. The mathematical perception of the gradient is said to be

16. Divergence of gradient of a vector function is equivalent to

17. Find the gradient of t = x2y+ ez at the point p(1,5,-2)

18. Curl of gradient of a vector is

19. The gradient can be replaced by which of the following?

21. The divergence of a vector is a scalar. State True/False.

Alternate/Shortcut: Without calculation, we can easily choose option b, as by

25. The curl of curl of a vector is given by,

26. Which of the following theorem use the curl operation?

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