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

And, from [Fig 1.46], we have the moments due to force ‘F ’ to be:

y

[Fig 1.46: Moments due to force 'Fz']

Moment due to F at a distance r = r i × F k = r F −j
z
z
x
x
x z

Moment due to F at a distance r = r j × F k = r F i

y
y
y z
z
z
Collecting and adding the F , F and F terms, we get:
x
y
z

F r j − r k + F r k − r i + F r i − r j - - - - (1.35)
y
x
z
z
y
x
x
z
y
Let us now cross verify equation (1.35) using the matrix form of vector product, as follows:

i j k
det r x r y r
z
F x F y F z

= i r F − r F − j r F − r F + k r F − r F
z y
z x
y z
x y
x z
y x

= i r F − i r F − j r F + j r F + kr F − kr F
x y
y x
y z
z x
z y
x z
Collecting the F , F and F terms, we get:
z
x
y

F r j − r k + F r k − r i + F r i − r j - - - - (1.36)
y
x
x
z
y
z
x
y
z

QP No. SSC/Q4401, Version 1.0, NSQF Level 7, Compliant with Aero and Auto Industries, Page 41