SLIDE 1 Rectangular components
y x z
F r i Fxˆ j Fy ˆ k Fz ˆ k ˆ j ˆ i ˆ
z
θ
x
θ
y
θ
2 2 2 z y x
F F F F + + = ) cos(
x x
F F θ =
) cos(
y y
F F θ = ) cos(
z z
F F θ =
k F j F i F F
z y x
ˆ ˆ ˆ + + = r ) ˆ cos ˆ cos ˆ (cos k j i F F
z y x
θ θ θ + + = r ) ˆ ˆ ˆ ( k n j m i l F F + + = r
F
n F F ˆ = r
x
l θ cos =
y
m θ cos =
z
n θ cos = Direction cosine 1
2 2 2
= + + n m l
SLIDE 2 Writing a vector in 3D (1)
1 Specification by two points on the line of action
x y z A (x1, y1, z1)
F r
B (x2, y2, z2)
A B AB
r r r r r r − = k z j y i x rA ˆ ˆ ˆ
1 1 1
+ + = r k z j y i x rB ˆ ˆ ˆ
2 2 2
+ + = r
| | ˆ
AB AB F
r r F n F F r r r = =
k z z j y y i x x rAB ˆ ) ( ˆ ) ( ˆ ) (
1 2 1 2 1 2
− + − + − = r
2 1 2 2 1 2 2 1 2 1 2 1 2 1 2
) ( ) ( ) ( ˆ ) ( ˆ ) ( ˆ ) ( z z y y x x k z z j y y i x x F F − + − + − − + − + − = r
SLIDE 3 Writing a vector in 3D (2)
2 Specification by two angles which orient the line of action
y x z
F r
x
F r
y
F r
z
F r
k ˆ j ˆ i ˆ
θ φ
xy
F r
Fxy = F cos(φ) Fz = F sin(φ) Fx = Fxy cos(θ) = F cos(φ) cos(θ) Fy = Fxy sin(θ) = F cos(φ) sin(θ)
SLIDE 4 Dot product
(scalar) P r Q r
α
Pcos(α) Projection of in the direction of P r Q r . Q ) cos(α PQ Q P = ⋅ r r
F r n r n F Fn ˆ ⋅ = r n n F Fn ˆ ˆ ⋅ = r r
If is a unit vector , dot product expresses the projection of vector in the unit vector direction Q r ) ˆ (n ) ˆ ˆ ˆ ( ) ˆ ˆ ˆ ( ˆ k j i k n j m i l F n F Fn γ β α + + ⋅ + + = ⋅ = r ) ( γ β α n m l F + + = 1 ˆ ˆ ˆ ˆ ˆ ˆ = ⋅ = ⋅ = ⋅ k k j j i i ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ = ⋅ = ⋅ = ⋅ = ⋅ = ⋅ = ⋅ j k k j i k k i i j j i
SLIDE 5 Angle between two vectors
) cos(θ PQ Q P = ⋅ r r From relation of dot product The angle between vectors and is P r Q r PQ Q P r r ⋅ =
−1
cos θ The angle between vectors and is F r n ˆ F n F ˆ cos 1 ⋅ =
−
r θ ˆ = ⋅n F r n F ˆ ⊥ r
SLIDE 6
Sample 1
A force F with a magnitude of 100 N is applied at the origin O of the axes x-y-z as shown. The line of action of F passes through a point A whose coordinates are 3m, 4m and 5m. Determine (a) the x, y and z scalar components of F, (b) the projection Fxy of F on the x-y plane, and (c) the projection FOB of F along the line OB.
SLIDE 7
Sample 2
The cable BC carries a tension of 750 N. Write this tension as a force T acting on point B in terms of the unit vector i, j and k. The elbow at A forms a right angle.
SLIDE 8
Sample 3
The tension in supporting cable BC is 3200 N. Write the force which this cable exerts on the boom OAB as a vector T. Determine the angles θx, θy and θz which the line of action of T forms with the positive x-, y- and z-axes.
