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Math Help - Vector and trigonometry-confusing

  1. #1
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    Vector and trigonometry-confusing

    Find the intersection of the lines r=-i+s(cosai+sinaj) and r=i+t(-sinai+cosaj), giving your answer in a simplified form. Interpret your answer geometrically.
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  2. #2
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    Solution

    This problem can be solved by 2 methods.
    Method 1:-
    r = -i + s(cosai + sinaj)
    or, r = (scosa - 1)i + ssinaj------I

    r = i + t(-sinai + cosaj)
    or, r = (1 - tsina)i + tcosaj-------II

    At the pt. of intersection of st. lines I & II,
    (scosa - 1)i + ssinaj = (1 - tsina)i + tcosaj
    =>scosa - 1 = 1 - tsina-------III
    & ssina = tcosa-------IV

    IV=> t = stana-------V
    III & V=> scosa - 1 = 1 - s(sin a)^2 * seca
    => s[cosa + (sina)^2/cosa] = 2
    => s = 2cosa-------VI

    V & VI=>
    t = 2cosa*tana
    => t = 2sina--------VII

    I & VI=>
    r = [2(cosa)^2 -1]i + 2cosa*sinaj
    => r = cos2ai + sin2aj--------VIII
    Therefore, the pt. of intersection is (cos2a,sin2a).

    Method 2:-
    Put r = xi +yj in both the eqs.
    Then, equating the coefficients of i & j for both the eqs. & solving them, we get the eqs. of the 2 st. lines as follows:-
    y = tana(x+1)-----A
    & y = cota(1-x)-----B

    Then, their pt. of intersection can be easily found out.
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