Lines in 3d
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Linear motion
Good way to learn about lines in \(\mathbb{R}^3\) is by studying trajectories of uniformly moving objects.
This 3d illustration shows a line with its direction vector \(\mathbf{v}\) through points \(P\) and \(Q\), together with positions corresponding to times \(t = -1\), \(t=0\), \(t=\frac{2}{3}\), \(t=1\) and \(t=2\):
Equations of lines
From the idea of uniform motion of an object in \(\mathbb{R}^3\), we obtain the vector equation of a line:
\[\mathbf{r}(t) = \mathbf{r}_0 + t\mathbf{v}\]
where \(\mathbf{r}(t)\) is the position vector of the object at time \(t\), \(\mathbf{r}_0\) is the initial position of the object (in other words, the position at \(t=0\)), and \(\mathbf{v}\) is the velocity vector of the moving object.
If we split this equation into individual components, we get the parametric equations of the line:
\[ \left\{ \begin{aligned} x(t) &= x_0 + v_1t\\ y(t) &= y_0 + v_2t\\ z(t) &= z_0 + v_3t \end{aligned} \right. \]
where \(x_0\), \(y_0\) and \(z_0\) are the components of the initial position vector \(\mathbf{r}_0 = \v{x_0, y_0, z_0\}\).
Finally, we can solve for \(t\) to get the symmetric equation of the line:
Try it yourself
Direction vector
The direction (or velocity) vector will tell us the direction of the line. If the direction vectors of two lines are parallel, then the lines are either parallel, or they are two different equations of the same line.
The components of the direction vector can tell us something about the line as well. For example:
If \(v_1 = 0\), the line’s distance from the \(yz\)-plane is constant, in other words, the line is parallel to the \(yz\)-plane. If \(v_1 \neq 0\), the line is not parallel to the \(yz\)-plane, and therefore must intersect it.
If \(v_2 = 0\), the line’s distance from the \(xz\)-plane is constant, in other words, the line is parallel to the \(xz\)-plane. If \(v_2 \neq 0\), the line is not parallel to the \(xz\)-plane, and therefore must intersect it.
If \(v_3 = 0\), the line’s distance from the \(xy\)-plane is constant, in other words, the line is parallel to the \(xy\)-plane. If \(v_3 \neq 0\), the line is not parallel to the \(xy\)-plane, and therefore must intersect it.
Intersection of lines
Two examples of equations of intersecting lines:
Examples of different ways how two lines can exist in \(\mathbb{R}^3\):
Illustrations of intersecting and non-intersecting lines in \(\mathbb{R}^3\):