Vector problem:

**karldiesen** · November 28th 2009, 07:19 AM

"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?

**Raoh** · November 28th 2009, 08:12 AM

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
$\begin{pmatrix} 1\\ 2\\ -3 \end{pmatrix}$ $,\begin{pmatrix} 2\\ 2\\ 3 \end{pmatrix}$ $,\begin{pmatrix} 1\\ 2-t\\ t+1 \end{pmatrix}$ $\Rightarrow$
$\begin{pmatrix} 1\\ 2\\ -3 \end{pmatrix}$ $,\begin{pmatrix} 0\\ -2\\ 9 \end{pmatrix}$ $,\begin{pmatrix} 0\\ 0\\ \frac{-7t+8}{2} \end{pmatrix}$
for those three victors to be coplanar we must have $rang(\begin{pmatrix} 1\\ 2\\ -3 \end{pmatrix}$ $,\begin{pmatrix} 0\\ -2\\ 9 \end{pmatrix}$ , $\begin{pmatrix} 0\\ 0\\ \frac{-7t+8}{2} \end{pmatrix} )$ $=2$ that is $\frac{-7t+8}{2}=0$ which means $t=\frac{7}{8}$
hope that's right

**Raoh** · November 28th 2009, 08:34 AM

Originally Posted by Raoh

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
$\begin{pmatrix} 1\\ 2\\ -3 \end{pmatrix}$ $,\begin{pmatrix} 2\\ 2\\ 3 \end{pmatrix}$ $,\begin{pmatrix} 1\\ 2-t\\ t+1 \end{pmatrix}$ $\Rightarrow$
$\begin{pmatrix} 1\\ 2\\ -3 \end{pmatrix}$ $,\begin{pmatrix} 0\\ -2\\ 9 \end{pmatrix}$ $,\begin{pmatrix} 0\\ 0\\ \frac{-7t+8}{2} \end{pmatrix}$
for those three victors to be coplanar we must have $rang(\begin{pmatrix} 1\\ 2\\ -3 \end{pmatrix}$ $,\begin{pmatrix} 0\\ -2\\ 9 \end{pmatrix}$ , $\begin{pmatrix} 0\\ 0\\ \frac{-7t+8}{2} \end{pmatrix} )$ $=2$ 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

)

**karldiesen** · November 28th 2009, 09:27 AM

Thanks for the answer, but I was hoping for a more "high-school level" answer!

**Plato** · November 28th 2009, 09:42 AM

Originally Posted by karldiesen

"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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