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 $v=(1,2,-3),u=(2,2,3),w=(1,2-t,t+1)$
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

3. 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 )

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