Which of the following is a true statement about the lines ewline $$oldsymbol{l_1 = egin{pmatrix} -1 \ 2 \ 5 \\ extcolor{red}{+} \lambda \begin{pmatrix} 3 \\ 1 \\ -1 \\ extcolor{red}{ ext{}} \\ extcolor{red}{ ext{}} \\ extcolor{red}{ ext{}} \\ extcolor{red}{ ext{}} \\ extcolor{red}{ ext{}} \\ extcolor{red}{ ext{}} \\ extcolor{red}{ ext{}} \\ extcolor{red}{ ext{}} \\ extcolor{red}{ ext{}} \\ extcolor{red}{ ext{}} \\ extcolor{red}{ ext{}} \\ extcolor{red}{ ext{}} \ 3} \\ -10} \ 1} \\ -3} \ 1}$$ ewline? - HSC - SSCE Mathematics Extension 2 - Question 5 - 2023 - Paper 1

Lines are parallel if their direction vectors are scalar multiples of each other. To check if $\mathbf{d_1}$ and $\mathbf{d_2}$ are parallel, we verify if there exists a scalar $k$ such that: $\mathbf{d_1} = k\mathbf{d_2}$

If we set up the equations from the components:

1. $3 = k(1)$
2. $1 = k(-3)$
3. $-1 = k(1)$

From the first equation, $k = 3$. However, substituting $k = 3$ into the second equation gives us

$1 \neq 3(-3) \implies 1 \neq -9$.

Thus, the lines are not parallel.

To determine if the lines intersect, we need to solve the equations:

1. $\begin{pmatrix} -1 + 3\lambda \end{pmatrix} = \begin{pmatrix} 3 \end{pmatrix}$
2. $\begin{pmatrix} 2 + \lambda \end{pmatrix} = \begin{pmatrix} -10 + \mu(-3) \end{pmatrix}$
3. $\begin{pmatrix} 5 + \lambda \end{pmatrix} = \begin{pmatrix} 1 \end{pmatrix}$

By solving these equations for $\lambda$ and $\mu$, we find that no solution exists. Thus, the lines are not the same and do not intersect.

Since the lines are neither parallel nor do they intersect, the correct statement is:

D. $l_1$ and $l_2$ are not parallel and they do not intersect.