Find a vector parametric equation $r⃗(t)$ for the line through the points $P=(3,5,4)$ and $Q=(1,4,7)$ for each of the given conditions on the parameter $t$.
If $r⃗ (0)=(3,5,4)$ and $r⃗ (7)=(1,4,7)$, then $r⃗ (t)= $?
I found the vector for $PQ$ and got $(-2,-1,3)$.
I then took the original $P$ and made $r(t) = (3,5,4) + t (-2,-1,3)$.
Not sure if I am just not really understanding the idea, but does anyone know how to solve this?
Best Answer
The idea here is that $\vec{r}(t)$ will reach $P$ at $t=0$ and $Q$ at $t=7$. Your parametric equation of $\vec{r}(t)$ is good, but what you need to do next is scale the velocity vector $\langle-2,1,3\rangle$ by a factor $x$ such that $\vec{r}(t)=P$ for $t=0$ and $Q$ for $t=7$. Following this logic, we have \begin{align*} (3,5,4)&=(3,5,4)+0x\langle -2, -1, 3\rangle \\ (1,4,7)&=(3,5,4)+7x\langle-2,-1,3\rangle \end{align*} which can be satisfied by $\displaystyle x=\frac{1}{7}$. So what is the intuition behind what we just did? You acquired $\langle -2, -1 3 \rangle$ by taking the difference of $P$ and $Q$, but we wanted $\vec{r}(t)$ to reach $Q$ at $t=7$ and not $t=1$, so we include the scale factor to compensate.