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Math Help - Vector problem:

  1. #1
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    Vector problem:

    "The vectors v = i + 2j - 3k, u = 2i + 2j + 3k, and w = i + (2 - t)j + (t + 1)k
    are given. Find the value of t such that the three vectors u, v and w are coplanar. "



    I have no idea how to solve this one!Where do I start?
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  2. #2
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    Smile

    let's take v=(1,2,-3),u=(2,2,3),w=(1,2-t,t+1)
    Using Gauss method on vectors,we should have
    \begin{pmatrix}<br />
1\\ <br />
2\\ <br />
-3<br />
\end{pmatrix} ,\begin{pmatrix}<br />
2\\ <br />
2\\ <br />
3<br />
\end{pmatrix} ,\begin{pmatrix}<br />
1\\ <br />
2-t\\ <br />
t+1<br />
\end{pmatrix} \Rightarrow
    \begin{pmatrix}<br />
1\\ <br />
2\\ <br />
-3<br />
\end{pmatrix} ,\begin{pmatrix}<br />
0\\ <br />
-2\\ <br />
9<br />
\end{pmatrix} ,\begin{pmatrix}<br />
0\\ <br />
0\\ <br />
\frac{-7t+8}{2}<br />
\end{pmatrix}
    for those three victors to be coplanar we must have  rang(\begin{pmatrix}<br />
1\\ <br />
2\\ <br />
-3<br />
\end{pmatrix} ,\begin{pmatrix}<br />
0\\ <br />
-2\\ <br />
9<br />
\end{pmatrix}, \begin{pmatrix}<br />
0\\ <br />
0\\ <br />
\frac{-7t+8}{2}<br />
\end{pmatrix}   ) =2 that is \frac{-7t+8}{2}=0 which means t=\frac{7}{8}
    hope that's right
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  3. #3
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    Smile

    Quote Originally Posted by Raoh View Post
    let's take v=(1,2,-3),u=(2,2,3),w=(1,2-t,t+1)
    Using Gauss method on vectors,we should have
    \begin{pmatrix}<br />
1\\ <br />
2\\ <br />
-3<br />
\end{pmatrix} ,\begin{pmatrix}<br />
2\\ <br />
2\\ <br />
3<br />
\end{pmatrix} ,\begin{pmatrix}<br />
1\\ <br />
2-t\\ <br />
t+1<br />
\end{pmatrix} \Rightarrow
    \begin{pmatrix}<br />
1\\ <br />
2\\ <br />
-3<br />
\end{pmatrix} ,\begin{pmatrix}<br />
0\\ <br />
-2\\ <br />
9<br />
\end{pmatrix} ,\begin{pmatrix}<br />
0\\ <br />
0\\ <br />
\frac{-7t+8}{2}<br />
\end{pmatrix}
    for those three victors to be coplanar we must have  rang(\begin{pmatrix}<br />
1\\ <br />
2\\ <br />
-3<br />
\end{pmatrix} ,\begin{pmatrix}<br />
0\\ <br />
-2\\ <br />
9<br />
\end{pmatrix}, \begin{pmatrix}<br />
0\\ <br />
0\\ <br />
\frac{-7t+8}{2}<br />
\end{pmatrix}   ) =2 that is \frac{-7t+8}{2}=0 which means t=\frac{7}{8}
    hope that's right
    in any case,u should wait until someone else confirm my post or the value of t.
    (i don't trust my answer anyway )
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  4. #4
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    Thanks for the answer, but I was hoping for a more "high-school level" answer!
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  5. #5
    MHF Contributor

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    Quote Originally Posted by karldiesen View Post
    "The vectors v = i + 2j - 3k, u = 2i + 2j + 3k, and w = i + (2 - t)j + (t + 1)k
    are given. Find the value of t such that the three vectors u, v and w are coplanar. "
    Here is another way.
    Solve W\cdot(V\times U)=0.
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