Which of the following is a true statement about the lines $$ egin{pmatrix} -1 \\ 2 \\ 5 ight) + \lambda egin{pmatrix} 3 \\ 1 \\ -1 ight) $$ and $$ egin{pmatrix} 3 \\ -10 \\ 1 \\ -3 \\ -1 ight) + \mu egin{pmatrix} -1 \\ -1 \\ -1 ight) $$ A - HSC - SSCE Mathematics Extension 2 - Question 5 - 2023 - Paper 1

The vector equation for line \ell_2 is given by:

-10 \\ 1\end{pmatrix} + \mu \begin{pmatrix}-1 \\ -1 \\ -1\end{pmatrix}$$ This results in the parametric equations: $$x = 3 - \mu,\ y = -10 - \mu,\ z = 1 - \mu$$

To determine if \ell_1$and$\ell_2$ are parallel, we compare their direction vectors: \begin{align*} \text{Direction of } \ell_1 &= \begin{pmatrix}3 \ 1 \ -1\end{pmatrix} \ \text{Direction of } \ell_2 &= \begin{pmatrix}-1 \ -1 \ -1\end{pmatrix} \end{align*}

Since the direction vectors are not scalar multiples of each other, the lines are not parallel.

Next, we set the parametric equations equal to each other to check for intersection: \begin{align*} -1 + 3\lambda &= 3 - \mu \ 2 + \lambda &= -10 - \mu \ 5 - \lambda &= 1 - \mu \end{align*}

Solving these equations would yield whether an intersection exists. Given they are not parallel, they are likely to intersect.

Based on the analysis, \ell_1$and$\ell_2$ are not parallel and they do intersect. Therefore, the correct statement is:

B. $\ell_1$ and $\ell_2$ are not parallel and they intersect.