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

Using the point A as the initial position vector and ( \lambda ) as the parameter, the vector equation of the line can be expressed as: ( \mathbf{r} = \begin{pmatrix} 4 \ 2 \ 5 \end{pmatrix} + \lambda \begin{pmatrix} -6 \ 0 \ -4 \end{pmatrix} ).

To compare with the given options, we notice that we can express the direction vector in a different way for simpler representation. Reordering can lead us to use integers that represent the direction effectively.

Thus, we will factor out common elements or correctly represent this using chosen multiples as necessary.

Upon analyzing the options, we can see that option B presents a different form consistent with the direction vector found, specifically by adjusting the vector to match a valid common direction.

Thus, the correct answer is Option B: ( \mathbf{r} = \begin{pmatrix} 4 \ 2 \end{pmatrix} + \lambda \begin{pmatrix} 0 \ 3 \end{pmatrix} ).