Which of the following is a true statement about the lines $$oldsymbol{ ext{l}_1 = egin{pmatrix} -1 \ 2 \ 5 \\ \lambda \begin{pmatrix} 3 \ 1 \ -1 \\ \\ \end{pmatrix} \\ ext{and} \\ ext{l}_2 = egin{pmatrix} 3 \ -10 \ 1 \\ \mu\begin{pmatrix} -3 \ 1 \ -1 \\ \\ \end{pmatrix}}$$? - HSC - SSCE Mathematics Extension 2 - Question 5 - 2023 - Paper 1

Two lines are parallel if their direction vectors are scalar multiples of each other. In this case, we can note:

eq k egin{pmatrix} -3 \ 1 \ -1 \\ \end{pmatrix}$$ for any scalar $k$. Thus, the lines are not parallel.

To check if the lines intersect, we can write:

egin{pmatrix} -1 \ 2 \ 5 \\ \end{pmatrix} + ext{l} egin{pmatrix} 3 \ 1 \ -1 \\ \end{pmatrix} = egin{pmatrix} 3 \ -10 \ 1 \\ \end{pmatrix} + ext{m} egin{pmatrix} -3 \ 1 \ -1 \\ \end{pmatrix}. Solving this system shows that $ext{l}$ and $ext{m}$ can take values such that solutions exist, indicating the lines do intersect.

Since the lines are not parallel and they do intersect, the correct statement is:

A. $ext{l}_1$ and $ext{l}_2$ are the same line.