Parametric Equations of Lines

An equation of a line in 3-space can be represented in terms of a series of equations known as parametric equations.

Parametric Equations of a Line

Suppose that we have a line $L$ in 3-space that passes through the points $P_0(x_0, y_0, z_0$ and $P(x, y, z)$. The vector $\vec{P_0P}$ lies on $L$. Now let $\vec{v} = (a, b, c)$ be a vector that is parallel to $L$. If $t$ is a scalar that spans from negative infinity to positive infinity then $\vec{P_0P} = t\vec{v}$, that is $\vec{v}$ traces out $L$ as $t$ goes from $-\infty$ to $+\infty$.

Multiplying this out we obtain that:

\begin{align} (x - x_0, y - y_0, z - z_0) = t(a, b, c) \\ (x - x_0, y - y_0, z - z_0) = (at, bt, ct) \end{align}

Thus we obtain the equations $x = x_0 + at$, $y = y_0 + bt$, and $z = z_0 + ct$ as $(-\infty < t < \infty)$ known as parametric equations.

Example 1

What are the parametric equations for the line $L$ that passes through the points $P(3, 4, 5)$ and $Q(-1, 2, -1)$?

We must first find a vector $\vec{v}$ that is parallel to the line formed by the points $P$ and $Q$. This is rather simple as we can construct the vector $\vec{PQ} = (-4, -2, -6)$ which is parallel to the line. All we need now is a point on the line and this question gives us two, so let's arbitrarily use $P$. Thus we obtain the following parametric equations:

\begin{align} x = 3 - 4t \quad , \quad y = 4 - 2t \quad , \quad z = 5 - 6t \quad (\infty < t < \infty) \end{align}

Example 2

At what point does the line $L$, passing through the points $P(3, 4, 5)$ and $Q(-1, 2, -1)$ intersect the $xz$-plane?

We already have the parametric equations for $L$ from example 1, that is:

\begin{align} x = 3 - 4t \quad , \quad y = 4 - 2t \quad , \quad z = 5 - 6t \quad (\infty < t < \infty) \end{align}

If $L$ intersects the $xz$-plane, then $y = 0$. Therefore we can find the value of $t$ that makes $y = 0$ by substituting into the y-equation:

\begin{align} y = 4 - 2t \\ 0 = 4 - 2t \\ t = 2 \end{align}

Thus $L$ intersects the $xz$-plane when $t = 2$. We can then find the other coordinates of this point by substituting t = 2 back into our other parametric equations to get the coordinates $(-5, 0, -7)$.

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