Checking Parallelism with Direction Vectors

00:00 3 marks

Given the line \(r: \vec{x} = \begin{pmatrix} 2 \\ -3 \end{pmatrix} + t\begin{pmatrix} 4 \\ 5 \end{pmatrix}\) and the line \(s: \vec{x} = \begin{pmatrix} -1 \\ 1 \end{pmatrix} + u\begin{pmatrix} -8 \\ -10 \end{pmatrix}\), determine if the two lines are parallel. Justify your answer by comparing their direction vectors.

Hint Hint

Hint

To check for parallelism, determine if the direction vector of line \(s\) is a scalar multiple of the direction vector of line \(r\).

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Complete Answer

The lines are parallel because the direction vector of line \(s\) is a scalar multiple of the direction vector of line \(r\), specifically \(\begin{pmatrix} -8 \\ -10 \end{pmatrix} = -2\begin{pmatrix} 4 \\ 5 \end{pmatrix}\).

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