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Static Case Study for dot product

  1. Oct 12, 2008 #1
    1. The problem statement, all variables and given/known data
    * With an axis system oriented as shown, the position vectors of points A and B are
    rA = 175i + 0j + 0k m
    rB = 39i + 70j + 29k m

    * When the trolley is halfway between points A and B, the forces exerted on the trolley by the cables are
    F1 = -943.7i - 221.7j + 245.4k m
    F2 = -919.4i - 216.0j - 328.6k m
    What is the component of the force exerted on the trolley that is parallel to the line AB?

    For detail : https://ecourses.ou.edu/cgi-bin/ebook.cgi?doc=&topic=st&chap_sec=02.4&page=case_intro

    2. Relevant equations

    The dot product of two vectors A and B is defined as the product of the magnitudes A and B and the cosine of the angle θ between them:

    A • B = |A| |B| cosθ

    The dot product can also be calculated by

    A • B = Ax Bx + Ay By + Az Bz
    Using the dot product, the angle between two known vectors A and B, can be determined as

    θ = cos^-1 AB/AB

    If the direction of a line is defined by the unit vector u, then the scalar component of the vector A parallel to that line is given by

    A|| = A • u

    The vector component parallel to that line is given by

    A|| = (A • u) u

    For Detail : https://ecourses.ou.edu/cgi-bin/ebook.cgi?doc=&topic=st&chap_sec=02.4&page=theory

    3. The attempt at a solution
    The scalar component of the total force parallel to the direction uAB can be found by using the dot product,

    FT|| = FT • uAB

    = -1,863 (-0.8736) - 437.7 (0.4496) - 83.2 (0.1863)

    = 1,415 N

    Multiply the scalar component by the unit vector uAB to determine the vector component parallel to the line AB:

    FT|| = FT • uAB

    = -1,236i + 636.3j + 263.7k N

    = 1,415 N

    For detail : https://ecourses.ou.edu/cgi-bin/ebook.cgi?doc=&topic=st&chap_sec=02.4&page=case_sol

    My question
    How is to find out the 1,415 N for scalar component and multiply scalar component??

    1. The problem statement, all variables and given/known data

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
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