Which of the following is a vector equation of the line joining the points A(4, 2, 5) and B(-2, 2, 1)? - HSC - SSCE Mathematics Extension 2 - Question 3 - 2021 - Paper 1

The vector equation of the line can be expressed as: $extbf{r} = extbf{a} + extbf{b} \lambda$ where

• $extbf{a}$ is the position vector of point A,
• $extbf{b}$ is the direction vector, and
• $extlambda$ is a scalar parameter.

Using point A(4, 2, 5) as the position vector: $extbf{a} = \begin{pmatrix} 4 \ 2 \ 5 \end{pmatrix}$ And the direction vector: $extbf{b} = \begin{pmatrix} -6 \ 0 \ -4 \end{pmatrix}$ The vector equation becomes: $extbf{r} = \begin{pmatrix} 4 \ 2 \ 5 \end{pmatrix} + \lambda \begin{pmatrix} -6 \ 0 \ -4 \end{pmatrix}$

From the options provided:

• A: $\begin{pmatrix} 4 \ 2 \ 5 \end{pmatrix} + \lambda \begin{pmatrix} 1 \ 2 \ 3 \end{pmatrix}$
• B: $\begin{pmatrix} 4 \ 2 \ 5 \end{pmatrix} + \lambda \begin{pmatrix} 3 \ 0 \ 2 \end{pmatrix}$
• C: $\begin{pmatrix} 1 \ 2 \ 3 \end{pmatrix} + \lambda \begin{pmatrix} 4 \ 2 \ 5 \end{pmatrix}$
• D: $\begin{pmatrix} 0 \ 2 \ 5 \end{pmatrix} + \lambda \begin{pmatrix} 4 \ 2 \ 2 \end{pmatrix}$

The correct answer corresponding to our derived equation is: B