- #1

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## Homework Statement

Write the parametric and cartesian equations of the line passing through ##P = (-\frac{3}{10}, 0, \frac{1}{10})## and parallel to

$$r : \begin{cases}

10x + 4y - 3 = 0 \\

x + z = 0

\end{cases}$$

## Homework Equations

## The Attempt at a Solution

Since the two lines have to be parallel, that means that the two directional vectors are linearly dependent. So I first calculate the directional vector of ##r## as follow:

##\vec v_r = \begin{vmatrix}

\hat i & \hat j & \hat k \\

10 & 4 & 0 \\

1 & 0 & 1

\end{vmatrix} = (4, -10, -4)##

Now that I have the directional vector, I know that the directional vector of the line we are searching for (let's call it ##u##) has to be linearly dependent to it, so something like this: ##\vec v_u = \alpha(4, -10, -4) = (4\alpha, -10\alpha, -4\alpha)##

With this I have already an overview of the parametric form of ##u## using ##\vec v_u## and ##P##, which is:

$$u : \begin{cases}

x = 4\alpha - \frac{3}{10}t \\

y = -10\alpha \\

z = -4\alpha + \frac{1}{10}t

\end{cases}$$

I was thinking about substituting this three equations into ##r## and find ##\alpha## but I stopped because I was doing it not knowing why. If it's the right way, why should I substitute them? If not, then, what should I do after arriving at this point?