ME 221 Statics
Sections 2.6 – 2.8
ME221
Lecture 6
1
Announcements
•HW #2 due Wednesday 9/10
2.22, 2.23, 2.25, 2.27, 2.29, 2.32, 2.37, 2.45,
2.47 & 2.50
•Quiz #2 on Wednesday 9/10
•Exam # 1 will be on Wednesday 9/17
ME221
Lecture 6
2
•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
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Lecture 6
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Case 2: Two Directions Known
y
P=A+B
P
a
P = Pcosβ î + Psinβ ĵ
b
x
Write out unit vectors
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Lecture 6
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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.
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Lecture 6
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The example we used had numerical values:
P = 100 lb, a = 10º, b = 20º
Set up the system of equations to solve
93.97 = A + 0.866 B
x-components
34.20 = 0 A + 0.5 B
y-components
Solving yields: B = 68.4 lb and A = 34.7 lb
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Lecture 6
6
Linear Algebraic Systems
Write the x- and y-component equations in
matrix form as follows:
Solve with your calculator.
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Lecture 6
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Summary Sections 2.6 – 2.7
• 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
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Lecture 6
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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
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Lecture 6
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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
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Lecture 6
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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
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Lecture 6
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Writing the Components
The dot product between two vectors is:
.
·
Components of a vector may be easily found
.
And finally
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Lecture 6
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Applications
• Determine the angle between two arbitrary
vectors
·
• Components of a vector parallel and
perpendicular to a specific direction
||
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Lecture 6
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