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Fundamentals of Stress and Vibration 1. Mathematics for Structural mechanics
[A Practical guide for aspiring Designers / Analysts]

[Fig 1.28: Parallelogram law of Vectors]

From [Fig 1.28], the longer diagonal of the parallelogram is given by:

2
2
2
2

a + b cosθ + b sinθ = a + b cos θ + 2 a b cosθ + b sin θ

2
2
2
Applying the fact that sin θ + cos θ = 1) in the above equation, we get:
2
2
2

a + b + 2 a b cosθ - - - - (1.26)
2
From the above discussion, we have the magnitudes of vectors [ a ] and [ b ] to be:

a = a + a and b = b + b

2
2
2
2
x
y
x
y
Substituting the magnitudes of the vectors in equation (1.26), we get:
a + a + b + b + 2 a + a b + b cosθ - - - - (1.27)
2
2
2
2
2
2
2
2
x
y
x
y
y
y
x
x
Equating, expression (1.27) with (1.25), we have:
2
2
2
2
a + b + a + b + 2 a b + a b = a + a + b + b + 2 a + a b + b cosθ
2
2
2
2
2
2
2
2
y
y
x
y
y
x
x
x x
x
y y
y
y
x
x
Cancelling the common terms in the above expression, we have:
a b + a b
cosθ = x x y y
a + a b + b 2
2
2
2
x y x y
The denominator a + a b + b , is the product of magnitude of two vectors.
2
2
2
2
y
y
x
x
QP No. SSC/Q4401, Version 1.0, NSQF Level 7, Compliant with Aero and Auto Industries, Page 29