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Math Help - Vector Help

  1. #1
    Member helloying's Avatar
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    Vector Help

    I dont know how to do qn 7 and 8. pls help pls.Ignore the blue ink on the diagram.
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  2. #2
    MHF Contributor red_dog's Avatar
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    7)
    a) i) \overrightarrow{OD}=\frac{4}{5}\overrightarrow{OM}  =\frac{4}{5}(\overrightarrow{OA}+\overrightarrow{A  B})=

    =\frac{4}{5}\left(\overrightarrow{OA}+\frac{1}{2}\  overrightarrow{AB}\right)=\frac{4}{5}\left(\overri  ghtarrow{OA}+\frac{1}{2}\overrightarrow{OC}\right)  =

    =\frac{4}{5}\left(p+q+\frac{p-q}{2}\right)=\frac{2(3p+q)}{5}

    ii) \overrightarrow{DB}=\overrightarrow{DM}+\overright  arrow{MB}=\frac{1}{5}\overrightarrow{OM}+\frac{1}{  2}\overrightarrow{AB}=

    =\frac{1}{5}\left(\overrightarrow{OA}+\frac{1}{2}\  overrightarrow{OC}\right)+\frac{1}{2}\overrightarr  ow{OC}=


    =\frac{1}{5}+\left(p+q+\frac{p-q}{2}\right)+\frac{p-q}{2}=\frac{4p-2q}{5}

    iii) \overrightarrow{NB}=\overrightarrow{NA}+\overright  arrow{AB}=\frac{1}{3}\overrightarrow{OA}+\overrigh  tarrow{OC}=\frac{4p-2q}{3}

    b) \overrightarrow{DB}=\frac{3}{5}\overrightarrow{NB}

    \overrightarrow{DB}, \ \overrightarrow{NB} are colinear.
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  3. #3
    MHF Contributor red_dog's Avatar
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    8) \overrightarrow{OA}=\begin{pmatrix}p\\0\end{pmatri  x}, \ \overrightarrow{OC}=\begin{pmatrix}q\\r\end{pmatri  x}

    \overrightarrow{OB}=\overrightarrow{OA}+\overright  arrow{OC}=\begin{pmatrix}p+q\\r\end{pmatrix}

    \overrightarrow{AC}=\overrightarrow{OC}-\overrightarrow{OA}=\begin{pmatrix}p-q\\r\end{pmatrix}

    OB^2+AC^2=(p+q)^2+r^2+(q-p)^2+r^2=2(p^2+q^2+r^2)

    2(OA^2+OC^2)=2(q^2+r^2+r^2)

    Then, OB^2+AC^2=2(OA^2+OC^2)
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