Unformatted text preview: Module 3: More Vectors
August 30, 2010 Module Content:
1. Cartesian vectors are especially useful forms of vectors.
2. Many physical quan==es relevant to structural analysis can be formulated as vectors.
3. Dot and cross product opera=ons also have speciﬁc physical meanings in sta=cs. Module Reading, Problems, and Demo:
Reading: Chapter 2
Problems: Fundamental Problem 2.3 (Mastering), Prob. 2.4
Demo: none
Homework PlaTorm: hUp://www.masteringengineering.com
Course Blog: hUp://pages.shan=.virginia.edu/sta=cs2010 MAE 2300 Sta=cs © E. J. Berger, 2010 3
1 Theory: Cartesian Vectors
• it is oYen useful to deﬁne vectors based upon a Cartesian coordinate system (x,y,z) • as such, we need to think about how we would deﬁne quan==es in speciﬁc direc=ons (like the x,y,z components of a vector) • to achieve this, we use...”unit vectors”, which simply deﬁne the (x,y,z) direc=ons themselves MAE 2300 Sta=cs © E. J. Berger, 2010 3
2 Theory: Cartesian Vectors
• it is oYen useful to deﬁne vectors based upon a Cartesian coordinate system (x,y,z) • as such, we need to think about how we would deﬁne quan==es in speciﬁc direc=ons (like the x,y,z components of a vector) • to achieve this, we use...”unit vectors”, which simply deﬁne the (x,y,z) direc=ons themselves MAE 2300 Sta=cs © E. J. Berger, 2010 3
2 Theory: Cartesian Vectors
• it is oYen useful to deﬁne vectors based upon a Cartesian coordinate system (x,y,z) • as such, we need to think about how we would deﬁne quan==es in speciﬁc direc=ons (like the x,y,z components of a vector) • to achieve this, we use...”unit vectors”, which simply deﬁne the (x,y,z) direc=ons themselves F = Fx i + Fy j
MAE 2300 Sta=cs © E. J. Berger, 2010 3
2 Theory: Cartesian Vectors
• it is oYen useful to deﬁne vectors based upon a Cartesian coordinate system (x,y,z) • as such, we need to think about how we would deﬁne quan==es in speciﬁc direc=ons (like the x,y,z components of a vector) • to achieve this, we use...”unit vectors”, which simply deﬁne the (x,y,z) direc=ons themselves F = Fx i + Fy j
MAE 2300 Sta=cs © E. J. Berger, 2010 3
2 Theory: Vector Speciﬁcs MAE 2300 Sta=cs © E. J. Berger, 2010 3
3 Theory: Vector Speciﬁcs MAE 2300 Sta=cs © E. J. Berger, 2010 3
3 Theory: Vector Speciﬁcs MAE 2300 Sta=cs © E. J. Berger, 2010 3
3 Theory: Vector Speciﬁcs MAE 2300 Sta=cs © E. J. Berger, 2010 3
3 Theory: Unit Vectors MAE 2300 Sta=cs © E. J. Berger, 2010 3
4 Theory: Unit Vectors
• a unit vector is a vector with magnitude equal to 1 MAE 2300 Sta=cs © E. J. Berger, 2010 3
4 Theory: Unit Vectors
• a unit vector is a vector with magnitude equal to 1 • any vector can be expressed in terms of a unit vector, as follows:
• consider a vector A, with magnitude A • a unit vector, with magnitude 1 and the same direcon as A, is called uA • then, we can write: A = A u MAE 2300 Sta=cs © E. J. Berger, 2010 3
4 Theory: Unit Vectors
• a unit vector is a vector with magnitude equal to 1 • any vector can be expressed in terms of a unit vector, as follows:
• consider a vector A, with magnitude A • a unit vector, with magnitude 1 and the same direcon as A, is called uA • then, we can write: A = A u MAE 2300 Sta=cs © E. J. Berger, 2010 3
4 Theory: Unit Vectors
• a unit vector is a vector with magnitude equal to 1 • any vector can be expressed in terms of a unit vector, as follows:
• consider a vector A, with magnitude A • a unit vector, with magnitude 1 and the same direcon as A, is called uA • then, we can write: A = A u • we have already seen the three units vectors associated with Cartesian vectors:
• ux = i • uy = j • uz = k MAE 2300 Sta=cs © E. J. Berger, 2010 3
4 Theory: Unit Vectors
• a unit vector is a vector with magnitude equal to 1 • any vector can be expressed in terms of a unit vector, as follows:
• consider a vector A, with magnitude A • a unit vector, with magnitude 1 and the same direcon as A, is called uA • then, we can write: A = A u • we have already seen the three units vectors associated with Cartesian vectors:
• ux = i • uy = j • uz = k MAE 2300 Sta=cs © E. J. Berger, 2010 3
4 Theory: How do we Uni=ze a Vector?
• consider a vector: A = Ax i + Ay j + Az k
• then, the magnitude of the vector is: A = √Ax2 + Ay2 + Az2
• and we can therefore deﬁne a unit vector in the direc=on of A as: uA = A =
A (hint: this looks like the direc0onal cosines on the next slide...)
MAE 2300 Sta=cs © E. J. Berger, 2010 3
5 Theory: More Vector Expressions
• there are several more ways to express vectors that you should read about:
• direc=onal cosines (p. 44, eq. 2
7) MAE 2300 Sta=cs © E. J. Berger, 2010 3
6 Theory: More Vector Expressions
• there are several more ways to express vectors that you should read about:
• direc=onal cosines (p. 44, eq. 2
7) MAE 2300 Sta=cs © E. J. Berger, 2010 3
6 Theory: More Vector Expressions
• there are several more ways to express vectors that you should read about:
• direc=onal cosines (p. 44, eq. 2
7) MAE 2300 Sta=cs © E. J. Berger, 2010 3
6 Theory: Posi=on Vectors
• posi=on vectors are simply vectors which locate one point with respect to another in space • absolute posi=on vectors are referenced to the origin of the coordinate system: • rela=ve posi=on vectors deﬁne a vector between two points in space: MAE 2300 Sta=cs © E. J. Berger, 2010 3
7 Theory: Posi=on Vectors
• posi=on vectors are simply vectors which locate one point with respect to another in space • absolute posi=on vectors are referenced to the origin of the coordinate system: • rela=ve posi=on vectors deﬁne a vector between two points in space: MAE 2300 Sta=cs © E. J. Berger, 2010 3
7 Theory: Posi=on Vectors
• posi=on vectors are simply vectors which locate one point with respect to another in space • absolute posi=on vectors are referenced to the origin of the coordinate system: • rela=ve posi=on vectors deﬁne a vector between two points in space: MAE 2300 Sta=cs © E. J. Berger, 2010 3
7 Theory: Posi=on Vectors
• posi=on vectors are simply vectors which locate one point with respect to another in space • absolute posi=on vectors are referenced to the origin of the coordinate system: • rela=ve posi=on vectors deﬁne a vector between two points in space: r = rB  rA MAE 2300 Sta=cs © E. J. Berger, 2010 3
7 Applica=on: Posi=on Vectors
• posi=on vectors are oYen 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 3
8 Applica=on: Posi=on Vectors
• posi=on vectors are oYen 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 3
8 Applica=on: Posi=on Vectors
• posi=on vectors are oYen 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 3
8 Applica=on: Posi=on Vectors
• posi=on vectors are oYen useful for deﬁning the direc=on of a force along a line of ac=on, such as in a cable, chair, or rope: u MAE 2300 Sta=cs © E. J. Berger, 2010 3
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