ME 221 Statics
Lecture #3
Sections 2.6 – 2.8
ME 221
Lecture 3
1
Announcements
• HW #1 Due Today
• Quiz #1 - 15 minutes before the end of the lecture
• HW #2 due Friday 5/28
Ch 2: 23, 29, 32, 37, 47, 50, 61, 82, 105, 113
Ch 3: 1, 8, 11, 25, 35
• Quiz #2 on Monday, May 24
• Exam # 1 will be on Wednesday, June 2
ME 221
Lecture 3
2
Summary of Last Lecture
• Writing vector components in terms of base
vectors
• Generating a 3-D unit vector from any
given vector
ME 221
Lecture 3
3
•Resolving Vectors into Components
Using Angle Notation
•Nonorthogonal Bases; Linear Equations
• Resolving vectors onto nonorthogonal
directions
• Setting up and solving linear systems of
algebraic equations
ME 221
Lecture 3
4
Resolving vector into components using angle notation
y
Ay
A
a
O
z
Az
Ax
x
b
A=A sina sinb i+A cosa j+A sina cosb k
ME 221
Lecture 3
5
Vector Components in Nonorthogonal
Coordinate System
y
A: Using Trig (law of sines):
v
sin(90 + a + b ) sin(q y  b ) sin(q x  a )


A
Au
Av
Ay
A
Av
qy
b
Au  A
qy-b
qx
90+a+b
u
a Au
sin(90 + a + b )
sin(q x  a )
Av  A
sin(90 + a + b )
x
Ax
ME 221
sin(q y  b )
Lecture 3
6
B: Using Vector Addition
Case 1: One Base Vector Known
y
y
B
-A
P
P
a
b
A
x
x
When vector A is known, subtract A from P
B = P-A
ME 221
Lecture 3
7
Case 2: Two Directions Known
y
P=A+B
P
a
P = Pcosβ î + Psinβ ĵ
b
x
Write out unit vectors
ME 221
Lecture 3
8
Write the components of P:
and write the vector sum equation.
Next, write the x and y component equations:
x-components
y-components
Here, we have two equations in two
unknowns, A and B. Solve the equations.
ME 221
Lecture 3
9
For example, using the numerical values:
P = 100 lb, a = 10º, b = 20º
Set up the system of equations to solve:
P cos β = 93.97 = A + 0.866 B
P sin β = 34.20 = 0 A + 0.5 B
x-components
y-components
Solving yields: B = 68.4 lb and A = 34.7 lb
ME 221
Lecture 3
10
Linear Algebraic Systems
Write the x- and y-component equations in
matrix form as follows:
Solve with your calculator.
ME 221
Lecture 3
11
Summary
• Be able to resolve a vector onto nonorthogonal directions
• Write the matrix form of the x-, y-, and zcomponent equations
• Be able to solve a 2 x 2 and 3 x 3 system of
equations on your calculator
ME 221
Lecture 3
12
Multiplying Vectors
Section 2.8
There are three basic ways vectors are multiplied
– Scalar times a vector
– Scalar product
• Often called the “dot” product
– Cross or vector product
ME 221
Lecture 3
13
Dot Product
Consider two vectors A and B with included
angle q
A
q
B
By definition, the dot product is
A • B = |A| |B| cos q
ME 221
Lecture 3
14
Dot Product of Base Vectors
• Let A and B be the base vectors and we find
·
·
·
since q = 0, then cos q = 1
• Also note that
·
·
·
since q = 90°, then cos q = 0
ME 221
Lecture 3
15
Writing the Components
The dot product between two vectors is:
.
·
Components of a vector may be easily found
.
And finally
ME 221
.
.
·
Lecture 3
16
Applications
• Determine the angle between two arbitrary
vectors
·
• Components of a vector parallel and
perpendicular to a specific direction
||
ME 221
·
·
Lecture 3
17
Example Problem
ME 221
Lecture 3
18
Quiz #1
ME 221
Lecture 3
19