# Thread: Vector problem:

1. ## 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?

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

3. Originally Posted by Raoh
let's take $\displaystyle v=(1,2,-3),u=(2,2,3),w=(1,2-t,t+1)$
Using Gauss method on vectors,we should have
$\displaystyle \begin{pmatrix} 1\\ 2\\ -3 \end{pmatrix}$$\displaystyle ,\begin{pmatrix} 2\\ 2\\ 3 \end{pmatrix}$$\displaystyle ,\begin{pmatrix} 1\\ 2-t\\ t+1 \end{pmatrix}$ $\displaystyle \Rightarrow$
$\displaystyle \begin{pmatrix} 1\\ 2\\ -3 \end{pmatrix}$$\displaystyle ,\begin{pmatrix} 0\\ -2\\ 9 \end{pmatrix}$$\displaystyle ,\begin{pmatrix} 0\\ 0\\ \frac{-7t+8}{2} \end{pmatrix}$
for those three victors to be coplanar we must have $\displaystyle rang(\begin{pmatrix} 1\\ 2\\ -3 \end{pmatrix}$$\displaystyle ,\begin{pmatrix} 0\\ -2\\ 9 \end{pmatrix},\displaystyle \begin{pmatrix} 0\\ 0\\ \frac{-7t+8}{2} \end{pmatrix} )$$\displaystyle =2$ that is $\displaystyle \frac{-7t+8}{2}=0$ which means $\displaystyle t=\frac{7}{8}$
hope that's right
in any case,u should wait until someone else confirm my post or the value of $\displaystyle t$.
(i don't trust my answer anyway )

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

5. 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 $\displaystyle W\cdot(V\times U)=0$.