ME 201 Engineering Mechanics: Statics

1
ME 201Engineering Mechanics Statics

• Chapter 2 Part C
• 2.7 Position Vectors
• 2.8 Force Vector Directed along a Line
• 2.9 Dot Product

2
Position Vectors

• The position vector r is defined as a fixed
  vector which locates a point in space relative to
  another point
• Distance and direction between 2 points
• Useful in formulating force vectors and finding
  the moment of a force

3
Position Vectors

• Simple Case from the origin
• r x i y j z k

Z
r
z
y
Y
x
X
4
Position Vectors

• General Case any 2 points A B
• the i, j, k components of the position vector
  r may be formed by taking the coordinates of the
  head of vector B and subtracting from them the
  corresponding coordinates of the tail A.

5
Position Vectors

• General Case any 2 points A B
• rAB (xB-xA) i (yB-yA) j (zB-zA) k

Z
B
rAB
A
Y
X
6
Vector/Scalar Review

• VECTOR
• Unit Vector U
• Position Vector r
• Force Vector F, T
• Moment - M

• SCALAR
• Magnitude
• Length
• Component

7
Steps to Formulate a Force Vector

• Find Position Vector
• rAB (xB-xA) i (yB-yA) j (zB-zA) k
• Find Magnitude of Position Vector
• Find Unit Vector
• Find Force Vector

8
Example Problem
z

• Given
• F 70 lb.
• Find
• F in vector form

30 m
B
A
y
8 m
6 m
12 m
x
9
Example Problem Solution
z
A
30 m
B
y
8 m
6 m
12 m

• Solution
• 1-Position vector
• 2-Magnitude
• 3-Unit vector
• 4-Force vector
• 5-Direction Cosines

A(0,0,30) B(12,-8,6)
10
Dot Product

• Dot Product is a scalar operation used in
  projecting vectors onto a given axis
• A B A B cos ?
• Where 0º ? 180º

A
?
B
11
Dot ProductLaws of Operation

• Commutative Law
• A B B A
• Multiplication by Scalar
• a(A B) (aA) B A (aB) (A B)a
• Distributive Law
• A (B D) (A B) (A D)

12
Dot Productof Cartesian Vector

• i i (1) (1) cos 0 1
• i j (1) (1) cos 90 0
• i k (1) (1) cos 90 0
• j j (1) (1) cos 0 1
• j k (1) (1) cos 90 0
• k k (1) (1) cos 0 1

13
Dot ProductCartesian Vector Formulation

• A B (Axi Ayj Azk) (Bxi Byj Bzk)
• AxBx(i i) AxBy(i j) AxBz(i k)
• AyBx(j i) AyBy(j j) AyBz(j k)
• AzBx(k i) AzBy(k j) AzBz(k k)
• AxBx AyBy AzBz

14
Dot ProductCartesian Vector Formulation

• A B AxBx AyBy AzBz
• A B cos ?

15
Projection of a Vector onto a Given Axis

• The scalar projection of A along a line is
  determined from the dot product of A and the unit
  vector U which defines the direction of the line
• A U A U cos ?
• A cos ?
• AU

A
?
U
AU
16
Example Problem

• Given
• F 300 j N
• Find
• FAB

z
B
F
3 m
A
y
2 m
6 m
x
17
Example Problem Solution
z
(2,6,3)
B
F
(0,0,0)
3 m
A
y
2 m
6 m

• Solution

x
Should we need it in Cartesian Vector form