My Math Forum 3 algebra problems

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 February 5th, 2014, 08:24 AM #1 Newbie   Joined: Feb 2014 Posts: 5 Thanks: 0 3 algebra problems Hello everybody, I am Francesco and I would be very grateful if someone could show me the solution for those 3 problems. I have many of them and these are just an example for type, so if I understand I can do the rest of my own. I don't find on google an exact answer to my problems, but all the time something similar but never the same :S. Thank you in advance if you can help the fellow here. http://s14.postimg.org/nhghwq4jl/Scr...t_17_13_53.png http://s14.postimg.org/jmd3u5ldt/Scr...t_17_16_16.png http://s14.postimg.org/iy49b7mo1/Scr...t_17_17_57.png
 February 5th, 2014, 07:11 PM #2 Math Team   Joined: Sep 2007 Posts: 2,409 Thanks: 6 Re: 3 algebra problems The first problem asks you to find the intersection of two lines in $R^4$. The first line is through $\begin{pmatrix}4 \\ -4 \\ 1 \\ 0 \end{pmatrix}$ and $\begin{pmatrix}-1 \\ 3 \\ 1 \\1 \end{pmatrix}$. The vector from the first point to the second can be written as $\begin{pmatrix}-1- 4 \\ 3- (-4) \\ 1- 1 \\ 1- 0\end{pmatrix}= \begin{pmatrix}-5 \\ 7 \\ 0 \\ 1\end{pmatrix}$ so that each point on the line can be written as $\begin{pmatrix}-5 \\ 7 \\ 0 \\ 1\end{pmatrix}t + \begin{pmatrix}4 \\ -4 \\ 1 \\ 0 \end{pmatrix}= \begin{pmatrix}-5t \\ 7t \\ 0 \\ t\end{pmatrix} + \begin{pmatrix}4 \\ -4 \\ 1 \\ 0 \end{pmatrix}$ or, equivalently, $\begin{pmatrix}-5t+ 4 \\ 7t- 4 \\ 1 \\ t\end{pmatrix}$. Notice that when t= 0, this is the first point, $\begin{pmatrix}4 \\ -4 \\ 1 \\ 0 \end{pmatrix}$ and when t= 1, it is the second point, $\begin{pmatrix}-1 \\ 3 \\ 1 \\1 \end{pmatrix}$. Do the same for the second line, using a different "name" for the variable, s, say, instead of t. Set the corresponding coordinates equal and solve for s and t. You will have 4 equations to solve for two unknowns. That is more equation than unknowns because, in general, two lines in 4 dimensions do NOT intersect. However, here, you have one line through points A and B and the second line between A and C so the point where they intersect should be obvious!

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