{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Engineering Calculus Notes 120

# Engineering Calculus Notes 120 - displacements In e²ect we...

This preview shows page 1. Sign up to view the full content.

108 CHAPTER 1. COORDINATES AND VECTORS 4. A 3 × 3 determinant equals zero precisely if its rows are linearly dependent. For the first item, note first that interchanging the two edges of the base reverses the sign of its oriented area and hence the sign of its oriented volume; if the first row is interchanged with one of the other two, you should check that this also reversed the orientation. Once we have the first item, we can assume in the second item that we are scaling the first row, and and in the second that A and B agree except in their first row(s). The additivity and homogeneity in this case follows from the fact that the oriented volume equals the oriented area of the base dotted with the first row. Finally, the last item follows from noting that zero determinant implies zero volume, which means the “height” measured off a plane containing the base is zero. Rotations So far, the physical quantities we have associated with vectors—forces, velocities—concern
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: displacements . In e²ect, we have been talking about the motion o± individual points, or the abstraction o± such motion ±or larger bodies obtained by replacing each body with its center o± mass. However, a complete description o± the motion o± solid bodies also involves rotation . A rotation o± 3-space about the z-axis is most easily described in cylindrical coordinates: a point P with cylindrical coordinates ( r,θ,z ), under a counterclockwise rotation (seen ±rom above the xy-plane) by α radians does not change its r- or z- coordinates, but its θ- coordinate increases by α . Expressing this in rectangular coordinates, we see that the rotation about the z-axis by α radians counterclockwise (when seen ±rom above) moves the point with rectangular coordinates ( x,y,z ), where x = r cos θ y = r sin θ to the point x ( α ) = r cos( θ + α ) y ( α ) = r sin( θ + α ) z ( α ) = z....
View Full Document

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern