Unformatted text preview: Module 4: Yes, Even More Vectors
September 1, 2010 Module Content:
1. Dot and cross product opera=ons also have speciﬁc physical meanings in sta=cs.
2. Par=cle equilibrium analysis rests upon free body diagrams. Module Reading, Problems, and Demo:
Reading: Chapters 2, 3
Problems: Prob. 2.90 (posi=on vectors/coordinate angles), Fund. Prob. 2.29 (dot product)
Demo: none
Homework PlaWorm: hXp://www.masteringengineering.com
Course Blog: hXp://pages.shan=.virginia.edu/sta=cs2010 MAE 2300 Sta=cs © E. J. Berger, 2010 4
1 Applica'on: Posi'on Vectors
• posi=on vectors are oZen useful for deﬁning the direc=on of a force along a line of ac=on, such as in a cable, chair, or rope: MAE 2300 Sta=cs © E. J. Berger, 2010 4
2 Applica'on: Posi'on Vectors
• posi=on vectors are oZen useful for deﬁning the direc=on of a force along a line of ac=on, such as in a cable, chair, or rope: MAE 2300 Sta=cs © E. J. Berger, 2010 4
2 Applica'on: Posi'on Vectors
• posi=on vectors are oZen useful for deﬁning the direc=on of a force along a line of ac=on, such as in a cable, chair, or rope: MAE 2300 Sta=cs © E. J. Berger, 2010 4
2 Applica'on: Posi'on Vectors
• posi=on vectors are oZen useful for deﬁning the direc=on of a force along a line of ac=on, such as in a cable, chair, or rope: MAE 2300 Sta=cs © E. J. Berger, 2010 4
2 Applica'on: Posi'on Vectors
• posi=on vectors are oZen useful for deﬁning the direc=on of a force along a line of ac=on, such as in a cable, chair, or rope: = magnitude of the posi1on vector MAE 2300 Sta=cs © E. J. Berger, 2010 4
2 Theory: Dot Products
• the dot product opera=on is deﬁned as the product of the magnitudes of two vectors mul=plied by the cosine of the angle between the vectors • in plain English, by example: project a vector F in 2D space onto the x
axis • and in general: MAE 2300 Sta=cs © E. J. Berger, 2010 4
3 Theory: Dot Products
• the dot product opera=on is deﬁned as the product of the magnitudes of two vectors mul=plied by the cosine of the angle between the vectors • in plain English, by example: project a vector F in 2D space onto the x
axis • and in general: MAE 2300 Sta=cs © E. J. Berger, 2010 4
3 Theory: Dot Products
• the dot product opera=on is deﬁned as the product of the magnitudes of two vectors mul=plied by the cosine of the angle between the vectors • in plain English, by example: project a vector F in 2D space onto the x
axis • and in general: MAE 2300 Sta=cs © E. J. Berger, 2010 4
3 Theory: Dot Products
• the dot product opera=on is deﬁned as the product of the magnitudes of two vectors mul=plied by the cosine of the angle between the vectors • in plain English, by example: project a vector F in 2D space onto the x
axis • and in general: MAE 2300 Sta=cs © E. J. Berger, 2010 4
3 Remarks: Dot Products
• there are lots of interes=ng mathema=cal rules about dot products...read about them on pages 68
69 • the few key points to remember:
•
• the dot product of two vectors can always be determined from the following basic rela=ons: •
• dot products produce scalar quan==es (NOT vectors) and similarly for the other combina=ons
and there is also an inverse formula=on for the dot product which allows us to calculate the angle between two vectors: MAE 2300 Sta=cs © E. J. Berger, 2010 4
4 Remarks: Dot Products
• there are lots of interes=ng mathema=cal rules about dot products...read about them on pages 68
69 • the few key points to remember:
•
• the dot product of two vectors can always be determined from the following basic rela=ons: •
• dot products produce scalar quan==es (NOT vectors) and similarly for the other combina=ons
and there is also an inverse formula=on for the dot product which allows us to calculate the angle between two vectors: MAE 2300 Sta=cs © E. J. Berger, 2010 4
4 Remarks: Dot Products
• there are lots of interes=ng mathema=cal rules about dot products...read about them on pages 68
69 • the few key points to remember:
•
• the dot product of two vectors can always be determined from the following basic rela=ons: •
• dot products produce scalar quan==es (NOT vectors) and similarly for the other combina=ons
and there is also an inverse formula=on for the dot product which allows us to calculate the angle between two vectors: MAE 2300 Sta=cs © E. J. Berger, 2010 4
4 Introducing Chapter 3: Par'cle Equilibrium
• par=cle equilibrium can be expressed as a speciﬁc case of Newton’s Law: • note that this is a vector equa=on • the way we actually write this equa=on is an explicit func=on of how we draw the diagram represen=ng the forces ac=ng on the par=cle: this diagram is the free body diagram • the usual way we approach this is to:
1. draw the par=cle 2. deﬁne a coordinate system 3. draw the forces MAE 2300 Sta=cs © E. J. Berger, 2010 4
5 Let’s Examine FBDs Together... MAE 2300 Sta=cs © E. J. Berger, 2010 4
6 ...
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