Vector equation of intersecting lines. Answer check

**Craka** · October 26th 2008, 05:41 PM

The question asks for the vector equation for the line of intersection by the planes x+2y-2z=5 and 6x-3y+2z=8

Answer is given as
$ \vec r = \left\langle {\frac{{31}}{{15}},\frac{{22}}{{15}},0} \right\rangle + t\left\langle {2,14,15} \right\rangle $

But when I do the working

Labeling x+2y-2z=5 as equation 1
and 6x-3y+2z=8 as equation 2

Multiply equation 1 by 6 to get 6x+12y-12z=30 call this equation 3

equation 2 minus equation 3 to get

$ \begin{array}{l} 15y = 22 - 14z \\ y = \frac{{22}}{{15}} - \frac{{14}}{{15}}z \\ \end{array} $

Let z=t

$ y = \frac{{22}}{{15}} - \frac{{14}}{{15}}t $

Than from equation 1

$ \begin{array}{l} x + 2y - 2z = 5 \\ x = 5 - 2\left( {\frac{{22}}{{15}} - \frac{{14}}{{15}}t} \right) - 2t \\ x = 5 - \frac{{44}}{{15}} + \frac{{28}}{{15}}t - 2t \\ x = \frac{{31}}{{15}} - \frac{2}{{15}}t \\ \end{array} $

and so

$ \begin{array}{l} \vec r = \left\langle {\frac{{31}}{{15}},\frac{{22}}{{15}},0} \right\rangle + t\left\langle {\frac{{ - 2}}{{15}},\frac{{ - 14}}{{15}},1} \right\rangle \\ \vec r = \left\langle {\frac{{31}}{{15}},\frac{{22}}{{15}},0} \right\rangle + t\left\langle { - 2. - 14,15} \right\rangle \\ \end{array} $

Am I right with my negative signs in <-2,-14,15>

and the answer in the text of <2,14,15> is wrong?

**katastrofa_nadfioletu** · October 27th 2008, 02:55 PM

Well, You've made two mistakes while 'rearranging' equations. There is:

$ \begin{array}{l} 15y = 22 - 14z \\ y = \frac{{22}}{{15}} - \frac{{14}}{{15}}z \\ \end{array} $

and should be:
$ \begin{array}{l} 15y = 22 + 14z \\ y = \frac{{22}}{{15}} + \frac{{14}}{{15}}z \\ \end{array} $
and the next minus is here:

$ \begin{array}{l} x + 2y - 2z = 5 \\ x = 5 - 2\left( {\frac{{22}}{{15}} - \frac{{14}}{{15}}t} \right) - 2t \\ x = 5 - \frac{{44}}{{15}} + \frac{{28}}{{15}}t - 2t \\ x = \frac{{31}}{{15}} - \frac{2}{{15}}t \\ \end{array} $

whereas it should be
$ \begin{array}{l} x + 2y - 2z = 5 \\ x = 5 - 2\left( {\frac{{22}}{{15}} - \frac{{14}}{{15}}t} \right) + 2t \\ x = 5 - \frac{{44}}{{15}} + \frac{{28}}{{15}}t + 2t \\ x = \frac{{31}}{{15}} + \frac{2}{{15}}t \\ \end{array} $
So, if You take it into consideration, it occurs that the answer in the text is correct.

best regards!
katastrofa.nadfioletu

Thread: Vector equation of intersecting lines. Answer check

LinkBack

Thread Tools

Display

Vector equation of intersecting lines. Answer check

Similar Math Help Forum Discussions

Intersecting lines

2(cd) intersecting lines

Answer check (Vector space)

vector equations answer check

need help! intersecting lines

Search Tags