Rectangular Components in Space Direction Cosines 3-d Rectangular components

3-d Rectangular components
Direction Cosines
Rectangular Components in Space
r
• The vector F is
contained in the
plane OBAC.
r
• Resolve F into
horizontal and vertical
components.
Fy = F cosθ y
Fh = F sin θ y
• Resolve F h into
rectangular components
Fx = Fh cos φ
= F sin θ y cos φ
Fy = Fh sin φ
= F sin θ y sin φ
Rectangular Components in Space
Sample Problem 2.7
SOLUTION:
• Based on the relative locations of the
points A and B, determine the unit
vector pointing from A towards B.
Direction of the force is defined by
the location of two points,
M ( x1 , y1 , z1 ) and N ( x2 , y 2 , z 2 )
• Apply the unit vector to determine the
components of the force acting on A.
The tension in the guy wire is 2500 N.
Determine:
• Noting that the components of the unit
vector are the direction cosines for the
vector, calculate the corresponding
angles.
a) components Fx, Fy, Fz of the force
acting on the bolt at A,
b) the angles θx, θy, θz defining the
direction of the force
1
Sample Problem 2.7
SOLUTION:
• Determine the unit vector pointing from A
towards B.
r
r
r
AB = (− 40 m ) i + (80 m ) j + (30 m )k
AB =
(− 40 m )2 + (80 m )2 + (30 m )2
= 94.3 m
r
− 40 ⎞ r ⎛ 80 ⎞ r ⎛ 30 ⎞ r
⎟k
⎟j +⎜
⎟i + ⎜
⎝ 94.3 ⎠ ⎝ 94.3 ⎠ ⎝ 94.3 ⎠
r
r
r
= −0.424 i + 0.848 j + 0.318k
λ = ⎛⎜
Sample Problem 2.7
• Noting that the components of the unit vector are
the direction cosines for the vector, calculate the
corresponding angles.
r
r
r
r
λ = cos θ x i + cos θ y j + cos θ z k
r
r
r
= −0.424 i + 0.848 j + 0.318k
θ x = 115.1o
θ y = 32.0o
θ z = 71.5o
• Determine the components of the force.
r
r
F = Fλ
r
r
r
= (2500 N )(− 0.424 i + 0.848 j + 0.318k )
r
r
r
= (− 1060 N )i + (2120 N ) j + (795 N )k
3-D Equilibrium of Particles
r
∑F
Ext
=0
-Equilibrium equation
-Can be used to find unknown forces
-A vector equation equivalent to 3
scalar component equations.
Example. Cable AB is attached to the top
of the vertical 3-m post, and its tension is
50 kN. What are the tensions in cables
AO, AC, and AD.
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