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Math Help - Vectors in 3D question

  1. #1
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    Vectors in 3D question

    lines L1 and L2 have vector equations r=8i-j+3k+ λ(-4i +j)
    and r= -2i +8j - k +
    (i +3j - 2k) respectively. it then says show that L1 and L2 intersect and find the position vector of the point of intersection.
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  2. #2
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    Hello, kandyfloss!

    Lines L_1 and L_2 have vector equations: . \begin{Bmatrix}\vec r&=&8\vec i-\vec j+3\vec k+ \lambda\left(-4\vec i +\vec j\right) \\ \\[-3mm]<br />
\vec r &=& \text{-}2\vec i +8\vec j - \vec k + \mu\left(\vec i +3\vec j - 2\vec k\right) \end{Bmatrix}

    Show that L_1 and L_2 intersect and find the position vector of the point of intersection.
    The vector equations can be written:

    . L_1\!:\;\;\begin{Bmatrix}x &=& 8 - 4\lambda & [1] \\ y &=& \text{-}1 + \lambda & [2] \\ z &=& 3 & [3] \end{Bmatrix} . . . L_2\!:\;\;\begin{Bmatrix}x &=& \text{-}2 + \mu & [4] \\ y &=& 8 + 3\mu & [5] \\ z &=& \text{-}1 - 2\mu & [6] \end{Bmatrix}


    \begin{array}{ccccccccc}<br />
\text{Equate [1] and [4]:} & 8-4\lambda &=& \text{-}2 + \mu & \Rightarrow & 4\lambda + \mu &=& 10 & [7] \\<br />
\text{Equate [2] and [5]:} & \text{-}1 + \lambda &=& 8 + 3\mu & \Rightarrow & \lambda - 3\mu &=& 9 & [8] \\<br />
\text{Equate [3] and [6]:} & 3 &=& \text{-}1 - 2\mu & \Rightarrow & \text{-}2\mu &=& 4 & [9]\end{array}


    From [9], we have: . {\color{blue}\mu \:=\:-2}

    Substitute into [8]: . \lambda - 3(\text{-}2) \:=\:9 \quad\Rightarrow\quad {\color{blue}\lambda \:=\:3}


    Substitute \lambda = 3 into [1], [2] and [3]:

    . . L_1\!:\;\;\begin{Bmatrix}x &=& 8-4(3) &=& \text{-}4 \\ y &=& \text{-}1 + 3 &=& 2 \\ z &=& 3 &=& 3\end{Bmatrix}


    Substitute \mu = -2 into [4], [5] and [6]:

    . . L_2\!:\;\;\begin{Bmatrix}x &=& \text{-}2 - 2 &=& \text{-}4 \\ y &=& 2 + 3(-2) &=& 2 \\ z &=& \text{-}1 - 2(\text{-}2) &=& 3 \end{Bmatrix}



    Therefore, the lines intersect at: . P(\text{-}4,2,3)

    The position vector of P is: .  \vec p \:=\:\langle \text{-}4,2,3\rangle \;=\;\text{-}4\vec i + 2\vec j + 3\vec k


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  3. #3
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    Excellent explanation!
    thanks
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