1. ## Vector equation of intersecting lines. Answer check

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

$
\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?

2. 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!