# Math Help - Vector: Lines Intersect

1. ## Vector: Lines Intersect

Vectors have to be my weakest points. How would I do this question, will I need to find $s$ and $t$? Help would be appreciated.

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Q:

Referred to a fixed origin $O$, the lines $l_1$ and $l_2$ have equations $\bold r = 3 \bold i + 6 \bold j + \bold k + s (2 \bold i + 3 \bold j - \bold k)$ and $\bold r = 3 \bold i - \bold j + 4 \bold k + t ( \bold i -2 \bold j + \bold k)$ respectively, where $s$ and $t$ are scalar parameters.

Show that $l_1$ and $l_2$ intersect and determine the position vector of their point of intersection.

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Thanks in advance.

2. Rewrite the lines as: $l_1 = \left\{ \begin{array}{r}
3 + 2s \\
6 + 3s \\
1 - s \\
\end{array} \right.\quad ,\quad l_2 = \left\{ \begin{array}{r}
3 + t \\
- 1 - 2t \\
4 + t \\
\end{array} \right.$

Now just solve the system.
But make sure the the solution works for all three equations.