Question

Find a vector parametric equation r(t) for the line through the points P=(−4,2,4) and Q=(−4,−3,5) for each of the given conditions on the parameter t. (a) If r(0)=⟨−4,2,4) and r(2)=⟨−4,−3,5⟩, then r(t)= ......

Answer 1

The vector parametric equation of the line through the points P=$(−4,2,4)$ and $Q=$(−4,−3,5)$ is given by:

$$r(t) = (−4 + 2t, 2 + t, 4 + 3t)$$

Step-by-step explanation:

To find the vector parametric equation of the line, we need to find the difference between the two points and express it as a function of the parameter t. Let's call the difference vector $\mathbf{d} = P - Q$. We have:

$$\mathbf{d} = (−4, −3, 5) - (−4, 2, 4) = (−7, 1, 1)$$

Now, we can write the vector parametric equation of the line as:

$$r(t) = P + t\mathbf{d} = (−4, 2, 4) + t(-7, 1, 1) = (−4 + 2t, 2 + t, 4 + 3t)$$

Note that $r(0) = P$ and $r(2) = Q$, so the equation satisfies the given conditions.

Step-by-step explanation: