Equations of Lines in Three-Dimensional Space
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Equations of Lines in Three-Dimensional Space

We will now look at describing lines in $\mathbb{R}^3$. The main form of describing lines is with parametric equations which we will elaborate on.

 Definition: Let $L$ be a line in $\mathbb{R}^3$. Let $P_0(x_0, y_0, z_0)$ and $P(x, y, z)$ be points on $L$ with $\vec{r_0}$ being the position vector with terminal point $P_0$ and $\vec{r}$ being the position vector with terminal point $P$ and let $\vec{v} = (a, b, c)$ be any vector parallel to $L$. Then the Vector Parametric Equation of $L$ is $\vec{r} = \vec{r_0} + t\vec{v}$, $-\infty < t < \infty$.

Recall that the position vector $\vec{r} = (x, y, z)$ can also be seen as a point in space, and $\vec{r} = \vec{r_0} + t\vec{v}$ means that the point $(x, y, z)$ in space is determined by starting at the point from $\vec{r_0}$ and then moving $t$ times the parallel vector $\vec{v}$. For $-\infty < t < \infty$, $\vec{r} = \vec{r_0} + t\vec{v}$ traces the entire line $L$.

Alternatively, we could represent $\vec{r} = \vec{r_0} + t \vec{v}$ as the following set of parametric equations $\left\{\begin{matrix} x = x_0 + at \\ y = y_0 + bt\\ z = z_0 + ct \end{matrix}\right.$.

Furthermore, if $a, b, c \neq 0$, then we can also solve each of these parametric equations and write the line as:

(1)
\begin{align} \frac{x - x_0}{a} = \frac{y - y_0}{b} = \frac{z - z_0}{c} \end{align}

Example 1

Find parametric equations of the line that passes through $P(-3, -4, -7)$ and that is parallel to the vector $\vec{v} = (1, 4, 2)$.

Substituting these values directly into the general form for parametric equations we have that the parametric equations that describe this line in particular are $\left\{\begin{matrix} x = -3 + t \\ y = -4 + 4t\\ z = -7 + 2t \end{matrix}\right.$ for $-\infty < t < \infty$. Notice that for $t = 0$ we have that $(x, y, z) = (-3, -4, -7)$ and so these parametric equations form a line that passes through $P$.

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