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Math Help - Urgent!! triple product

  1. #1
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    Urgent!! triple product

    Show that the lines

    x=a1s+ b1, y= a2s + b2 , z =a3s+b3 , -infinity<s<infinity

    and


    x=c1t+ d1, y= c2t + d2 , z =c3t+d3 , -infinity<t<infinity

    intersect or parallel if and only if

    | a1 c1 b1-d1 |
    | a2 c2 b2-d2 | = 0
    | a3 c3 b3-d3 |
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  2. #2
    Senior Member Peritus's Avatar
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    1) Do not create identical threads. Do not double post.
    Please read the forum rules, it will increase your chance of getting help from the forum members.
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  3. #3
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    Hello, fatduck88!

    I have a start on this problem anyway . . .


    Show that the lines: . \begin{Bmatrix}x&=&a_1s+ b_1 \\y&=& a_2s + b_2 \\ z &=&a_3s+b_3\end{Bmatrix}\;\text{ and }\;\begin{Bmatrix}x&=&c_1t+ d_1 \\ y&=&c_2t + d_2 \\ z&=&c_3t+d_3 \end{Bmatrix}

    . . intersect or are parallel if and only if: . \begin{vmatrix}\:a_1 & c_1 &(b_1\!-\!d_1)\: \\ a_2 & c_2 & (b_2\!-\!d_2) \\ a_3 & c_3 & (b_3\!-\!d_3) \end{vmatrix} \;=\;0

    The lines intersect if: . \begin{array}{ccc}a_1s+b_1 &=& c_1t+d_1 \\ a_2s+b_2 &=& c_2t+d_2 \\ a_3s+b_3 &=&c_3t+d_3 \end{array}


    . . That is, if the system: . \begin{array}{ccc}a_1s - c_1t + (b_1-d_1) &=&0 \\ a_2s - c_2t + (b_2-d_2) &=& 0 \\ a_3s - c_3t + (b_3-d_3) &=&0 \end{array} . has a solution (s,t)


    Now we must relate this system to that determinant . . .


    [But I truly dislike "if and only if" problems . . . p'too!]
    .
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