2.11 Direction Cosines
17
Let
cos α = l, cos β = m and cos γ = n
Therefore,
x
r
= l,
y
r
= m and
z
r
= n
Here, l, m and n are known as direction cosines of the line OP. Also, it can be
written as
x
2
+ y
2
+ z
2
= r
2
(since r is the polar co-ordinate of P)
or
x
2
r 2 +
y
2
r 2 +
z
2
r 2 = 1
l
2
+ m
2
+ n
2
= 1 (This is well known in co-ordinate geometry)
2.12 Stress Components on an Arbitrary Plane
Consider a small tetrahedron isolated from a continuous medium (Fig. 2.9) subjected
to a general state of stress. The body forces are taken to be negligible. Let the arbitrary
plane ABC be identified by its outward normal n whose direction cosines are l, m
and n.
Fig. 2.9 Stresses acting on
face of the tetrahedron
17
Let
cos α = l, cos β = m and cos γ = n
Therefore,
x
r
= l,
y
r
= m and
z
r
= n
Here, l, m and n are known as direction cosines of the line OP. Also, it can be
written as
x
2
+ y
2
+ z
2
= r
2
(since r is the polar co-ordinate of P)
or
x
2
r 2 +
y
2
r 2 +
z
2
r 2 = 1
l
2
+ m
2
+ n
2
= 1 (This is well known in co-ordinate geometry)
2.12 Stress Components on an Arbitrary Plane
Consider a small tetrahedron isolated from a continuous medium (Fig. 2.9) subjected
to a general state of stress. The body forces are taken to be negligible. Let the arbitrary
plane ABC be identified by its outward normal n whose direction cosines are l, m
and n.
Fig. 2.9 Stresses acting on
face of the tetrahedron