Find a vector equation and parametric equations for the line segment that joins $ P $ to $ Q $.
$ P (-1, 2, -2), Q (-3, 5, 1) $
Where $t \in[0,1]$
So what we want to do to find the vector equation of a line segment joining two points are not in our one is are vector position Vector is equal to one minus t times. Our first are vector are not plus t times are one are not in Our one are going to be based off of our points. So we know that are not equals One negative 12 negative two And our one is going to be equal to negative 35 and one. So now we can plug in these components and we get that are not Our vector is equal to one minus t times negative 1 to 2 Victor plus t times negative 351 vector Then we multiply us out Combine like terms, um, and make sure we keep all the components together X, y and z we get that are vector the position vector as a function of T is equal to negative one minus two t two plus three t negative two plus three t where t is greater than or equal to zero but less than equal to one. And then we want to show the parametric equations that'll be negative on minus two. T for X Y is equal to two plus three. T n Z is equal to negative two plus three t and again this still applies.