Determine the point of intersection between the following pair of lines
\[
\begin{array}{l}
\vec{r}=(5,1,7)+s(2,0,5), s \in R \\
\vec{r}=(-1,-1,3)+t(4,2,-1), t \in R
\end{array}
\]

Answer

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Answer

\(\boxed{(3, 1, 2)}\)

Steps

Step 1 ：\(\begin{cases} 5 + 2s = -1 + 4t \\ 1 = -1 + 2t \\ 7 + 5s = 3 - t \end{cases}\)

Step 2 ：\(\begin{cases} 2s - 4t = -6 \\ 2t = 2 \\ 5s + t = -4 \end{cases}\)

Step 3 ：\(t = 1\)

Step 4 ：\(\begin{cases} 2s - 4(1) = -6 \\ 5s + 1 = -4 \end{cases}\)

Step 5 ：\(\begin{cases} 2s = -2 \\ 5s = -5 \end{cases}\)

Step 6 ：\(s = -1\)

Step 7 ：\(\vec{r}=(5,1,7)+(-1)(2,0,5) = (3,1,2)\)

Step 8 ：\(\boxed{(3, 1, 2)}\)