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

However, from elementary trigonometry, we know that, the area of a triangle whose sides and
included angle are known is given by:

1

Area of the triangle ACD = a b sinθ
2
This proves equation (1.30).

Vector products could also be represented in the matrix form, as follows:

i j k

a × b = a x a y a
z
b x b y b z
Taking determinant of the above matrix, we get:

k
i j
det a x a y a
z
b x b y b z

= i a b − a b − j a b − a b + k a b − a b
y z
y x
z y
z x
x z
x y
The above expression is the vector whose magnitude is given by: a b sinθ

Which is the area of the parallelogram, whose two sides the vectors a and b themselves.

Example 1: a force applied in a 3D space
has 3 components, one in each direction
(x, y, and z). The position vector ‘r’ also has
3 components (x, y, and z).

Let the 3 components of force be F , F and F
y
x
z
and the distances be r , r and r as
z
x
y
shown in [Fig 1.43].

Each component of force produces two
moments as one of the position vectors
coincides with its own direction, which
cannot produce a moment.

[Fig 1.43: Forces on a Point in a 3D space]

QP No. SSC/Q4401, Version 1.0, NSQF Level 7, Compliant with Aero and Auto Industries, Page 39