Question

find a vector function $mathbf{r}(t)$ that, as $t$ goes from $-infty$ to $infty$, will generate the entire line through the points $(-1, -7, -4)$ and $(-9, 7, 2)$. answer: $mathbf{r}(t)=$

Explanation:

Step1: Find the direction vector

Let $P(-1,-7,-4)$ and $Q(-9,7,2)$. The direction vector $\vec{v}=\langle -9 - (-1),7-(-7),2 - (-4)
angle=\langle - 8,14,6
angle$.

Step2: Write the vector - function formula

The vector - function of a line passing through the point $(x_0,y_0,z_0)$ with direction vector $\vec{v}=\langle a,b,c
angle$ is $\mathbf{r}(t)=\langle x_0 + at,y_0+bt,z_0 + ct
angle$. Here $(x_0,y_0,z_0)=(-1,-7,-4)$ and $\vec{v}=\langle - 8,14,6
angle$. So $\mathbf{r}(t)=\langle -1-8t,-7 + 14t,-4+6t
angle$.

Answer:

$\mathbf{r}(t)=\langle -1-8t,-7 + 14t,-4+6t
angle$