Engineering Calculus Notes 104

Engineering Calculus Notes 104 - segments to be the sum of...

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

View Full Document Right Arrow Icon
92 CHAPTER 1. COORDINATES AND VECTORS (b) Show that the above is not true if B is not between A and C . (c) Show that σ ( O ,A,B ) A ( △O AB ) + σ ( O ,B,C ) A ( △O BC ) + σ ( O ,C,A ) A ( △O CA ) = 0 regardless of the order of A , B and C along the line. 10. Show that the oriented area of a triangle can also be calculated as half of the cross product of the vectors obtained by moving along two successive edges: vector A ( ABC ) = 1 2 −−→ AB × −−→ BC ( Hint: You may use Exercise 8 .) Challenge Problems: Given a point D in the plane, and a directed line segment −−→ AB , we can define the area swept out by the line DP as P moves from A to B along −−→ AB to be the signed area of the oriented triangle [ D,A,B ]. We can then extend this definition to the area swept out by DP as P moves along any broken-line path ( i.e. , a path consisting of finitely many directed line
Image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: segments) to be the sum of the areas swept out over each of the segments making up the path. 11. (a) Show that the area swept out by DP as P travels along an oriented triangle equals the signed area of the triangle: that is, show that σ ( ABC ) A ( △ ABC ) = σ ( DAB ) A ( △ DAB )+ σ ( DBC ) A ( △ DBC )+ σ ( DCA ) A ( △ DCA ) . ( Hint: This can be done geometrically. Consider three cases: D lies outside, inside, or on △ ABC . See ±igure 1.47 .) (b) Show that the area swept out by O P as P moves along the line segment from ( x ,y ) to ( x 1 ,y 1 ) is 1 2 v v v v x y x 1 y 1 v v v v ....
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    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.

    Student Picture

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

  • Left Quote Icon

    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.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    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.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern