St_Mod3_F10

St_Mod3_F10 - Module 3: More Vectors August 30, 2010...

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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 direc-on 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 direc-on 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 direc-on 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 direc-on 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 ­ 8 ...
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This note was uploaded on 02/09/2012 for the course MAE 2300 taught by Professor Staff during the Fall '10 term at UVA.

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