The question is:
Find the vector equation of the line with Cartesian equation:
$$5x + 1 = -10y – 4 = 2z$$
I know the vector equation of a line is $\textbf{r} × \textbf{v} = \textbf{a} × \textbf{v}$,
where $\textbf{r}$ is the position vector of a point on the line, $\textbf{a}$ is a fixed point on
the line, and $\textbf{v}$ is a direction vector for $\textit{L}$. What I don't get is how the Cartesian equation can give me what I need.
If someone could please explain the process of converting Cartesian into vector form and highlight the link between Cartesian and vector equations of lines that would be great.
Best Answer
Hint:
You have: $$ 5x + 1 = -10y - 4 = 2z=t $$
so: $$ x=\frac{t-1}{5} \qquad y=\frac{t+4}{-10} \qquad z=\frac{t}{2} $$
can you find a vector equation from this?
(It has the form $\vec x= t \vec v+ \vec w$)