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Math Help - Vectors: unknown components

  1. #1
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    Unhappy Vectors: unknown components

    I have this question: "If vector c forms an obtuse angle with the OY-axis and is orthogonal to both vectors a = 3i + 2j + 2k and b = 18i - 22j - 5k. Find the component of c if it's norm is 14"

    This I can gather:

    a dot c = 0 (because they are orthogonal)
    b dot c = 0 (because they are orthogonal)
    a x b = 0 (because both a and b are orthogonal to c, a and b must be paralell)
    |a| = sqrt(17)
    |b| = 7sqrt(17)
    |a x c| = 119
    |a x b| 98sqrt(17)

    I know from the equation that if the angle is obtuse with the 0Y axis, the Y component must be negative, so I surmise that c = c1 *i - c2 *j + c3*k

    Other than that I'm completely lost, I can't figure out how to find the components of vector c
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  2. #2
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    Quote Originally Posted by veralix View Post
    I have this question: "If vector c forms an obtuse angle with the OY-axis and is orthogonal to both vectors a = 3i + 2j + 2k and b = 18i - 22j - 5k. Find the component of c if it's norm is 14"
    First \;b \times a = \left\langle { - 34, - 51, + 102} \right\rangle \;\& \;\left\| {b \times a} \right\| = 119 perpendicular to both making obtuse angle with j.
    Make a unit vector and multiply by 14: \left\langle {\frac{{ - 476}}<br />
{{119}},\frac{{ - 714}}<br />
{{119}},\frac{{1428}}<br />
{{119}}} \right\rangle \;
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  3. #3
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    Or:
    Writing c as c_1\vec{i}+ c_2\vec{j}+ c_3\vec{k}, knowing that \vec{a} and \vec{b} are orthogonal, you know that \vec{a}\cdot\vec{c}= 3c_1+ 2c_2+ 2c_2= 0 and knowing that \vec{b} and \vec{c} are othogonal, you know that 18c_1- 22c_2- 5c_3= 0. Knowing that |\vec{c}|= 14 you know that c_1^2+ c_2^2+ c_3^2= 196. You can solve the first two, linear, equations for c_1 and c_2 as functions of c_3 and put those into the last equation to get a quadratic equation for c_3. It's the fact that that quadratic equation has two solutions that requires that additiona information about the angle.
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