Solved Examples / Numericals On Vector Algebra - Field Theory.

Q.1) Given vectors A = a_x + 3a_z and B = 5a_x + 2a_y -6a_z, determine

a) |A+B|
b) 5A – B
c) Component of A along a_y.
d) Unit vector parallel to 3A + B.

Ans:

a) A + B = a_x +3a_z +5a_x +2a_y - 6a_z = 6a_x + 2a_y – 3a_z.

|A + B| = (6² + 2² + 3²)^1/2 = 7.


b) 5A – B = 5(a_x + 3a_z) – 5a_x – 2a_y + 6a_z = -2a_y + 21a_z.


c) Component of A along a_x, a_y and a_z are 1, 0 and 3 respectively.


d) 3A + B = 3a_x + 9a_z + 5a_x + 2a_y -6a_z = 8a_x + 2a_y +3a_z

a_{3A + B} = 3A + B/ |3A + B| = (8a_x + 2a_y + 3a_z)/ (77)^1/2


Q.2) Given Points P (1, -3, 5), Q (2, 4, 6) and R (0, 3, 8) find
a) Position vectors of P and R.
b) Distance vector r_QR.
c) Distance between Q and R.

Ans:

a) r_p = a_x – 3a_y + 5a_z.
r_R = 3a_y +8a_z.


b) r_QR = r_R -r_Q
= 3a_y +8a_z – (2a_x + 4a_y + 6a_z)
= - 2a_x – a_y + 2a_z.


c) |r_QR| = (2² + 1² + 2²)^1/2 = 3.


Q.3) Find the unit vector along the line joining point (2, 4, 4) to point (-3, 2, 2)?

Ans:

Ā = (-3, 2, 2) – (2, 4, 4)
= (-5, -2, -2)

= - 0.87a_x - 0.35a_y - 0.35a_z




Q.4) Given that A = 3a_x + 5a_y – 7a_z and B = a_x – 2a_y + a_z ; find

a) | 2B + 0.4A |
b) A.B - | B |²
c) A x B
 Ans:

a) 2B + 0.4A
= 2a_x – 4a_y +2a_z + 0.4 (3a_x + 5a_y – 7a_z)
= 3.2a_x + 6a_y - 0.8a_z

| 2B + 0.4A | = (3.2² + 6² + 0.8²)^1/2 = 6.846


b) A.B - | B |²
= (3a_x + 5a_y -7a_z) . (a_x -2a_y + a_z) – (1² + 2² +1²)^1/2
= - 14 – (6)^1/2
= - 16.4494


c) A x B

Q.5) Given Vectors T = 2a_x – 6a_y + 3a_z and S = a_x + 2a_y +az; find

a) the scalar projection of T on S.
b) the vector projection of S on T.
c) the smaller angle between T and S.
Ans:

a)

b)

= -0.286 a_x + 0.857a_y – 0.43a_z


c)

Sin θ_TS = 0.9129 => θ_TS = 65.91^o



Q.6) Let E = 3a_y + 4a_z and F = 4a_x – 10a_y + 5a_z

a) Find the component of E along F.
b) Determine a Unit vector perpendicular to both E and F.
Ans:

a)

= - 0.28a_x +0.71a_y - 0.35a_z

b)

= ± (0.94, 0.27, -0.21)


Q.7) E and F are vector fields given by E = 2xa_x + a_y +yza_z. and
F = xya_x – y2a_y +xyza_z. Determine:

a) | E| at (1, 2, 3)
b) Component of E along F at (1, 2, 3)
c) A vector perpendicular to both E and F at (0,1, -3) whose magnitude is unity?
Ans:


a) At (1, 2, 3), E = (2, 1, 6)

b) At (1, 2, 3),  F = (2, -4, 6)

= 1.29 a_x – 2.57 a_y + 3.86 a_z


c) At (0, 1, -3), E = (0, 1, -3) and F = (0, -1, 0)

ALSO READ:

- Scalars and Vectors.

- Unit Vectors.

- Position and Distance Vectors.

- Vector Multiplication.

- Components Of a Vector.

- Line , Surface and Volume Intergral.

- Del Operator - Definition and Significance.

- Gradient Of a Scalar (∇ V).

- Divergence Of a Vector ( ∇ . A ).

- Curl Of a Vector ( ∇ x A).

- Laplacian Of a Scalar ( ∇² V).


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