Section 12.5: Lines & Planes

Often, we need a description, not of an entire line, but of just a line segment. How, for instance, could we describe the line segment AB in Example 2? If we put \(t = 0\) in the parametric equations in Example 2(a), we get the point \((2, 4, -3)\) and if we put \(t = 1\) we get \((3, -1, 1)\). So the line segment AB is described by the parametric equations \[ x = 2 + t \quad y = 4 - 5t \quad z = -3 + 4t \quad 0 \le t \le 1 \] or by the corresponding vector equation \[ \mathbf{r}(t) = \langle 2 + t, 4 - 5t, -3 + 4t \rangle \quad 0 \le t \le 1 \] In general, we know from Equation 1 that the vector equation of a line through the (tip of the) vector \(\mathbf{r}_0\) in the direction of a vector v is \(\mathbf{r} = \mathbf{r}_0 + t\mathbf{v}\). If the line also passes through (the tip of) \(\mathbf{r}_1\), then we can take \(\mathbf{v} = \mathbf{r}_1 - \mathbf{r}_0\) and so its vector equation is \[ \mathbf{r} = \mathbf{r}_0 + t(\mathbf{r}_1 - \mathbf{r}_0) = (1 - t)\mathbf{r}_0 + t\mathbf{r}_1 \] The line segment from \(\mathbf{r}_0\) to \(\mathbf{r}_1\) is given by the parameter interval \(0 \le t \le 1\).

The line segment from \(\mathbf{r}_0\) to \(\mathbf{r}_1\) is given by the vector equation \[ \mathbf{r}(t) = (1 - t)\mathbf{r}_0 + t\mathbf{r}_1 \quad 0 \le t \le 1 \tag{4} \]