Hey guys, quick question. If a plane passes through the points (a,b,c) and (d,e,f), and is parallel to the vector (3,2,1). Am i right in assuming the vector equation for the plane will be (x,y,z) = (a,b,c) + [(d,e,f) - (a,b,c)]s + (3,2,1)s. Thanks
Hey guys, quick question. If a plane passes through the points (a,b,c) and (d,e,f), and is parallel to the vector (3,2,1). Am i right in assuming the vector equation for the plane will be (x,y,z) = (a,b,c) + [(d,e,f) - (a,b,c)]s + (3,2,1)s. Thanks
If it passes through points $\displaystyle (a,b,c)$ and $\displaystyle (d,e,f)$ then $\displaystyle t(a - d, b - e, c - f)$ is also a vector it is parallel to.
$\displaystyle (x,y,z) = (a,b,c) + t(a-d, b-e, c-f)$
$\displaystyle (x,y,z) = (a,b,c) + t(3, 2, 1)$
I suppose you could combine them into:
$\displaystyle 2(x,y,z) = 2(a,b,c) + t(3 + a - d, 2 + b - e, 1 + c - f)$
$\displaystyle (x,y,z) = (a,b,c) + \frac{t(3 + a - d, 2 + b - e, 1 + c - f)}{2}$
EDIT:
But I'm not sure why you would use such an elaborate method.
$\displaystyle (x,y,z) = (a,b,c) + t_1 (3,2,1)$
$\displaystyle (x,y,z) = (d,e,f) + t_2 (3,2,1)$
You could use either one of those equations as well. If this plane passes through both points, and is parallel to a certain vector then you only need one of those points and the vector it's parallel to.
Nearly! You have to use a 2nd variable. (see my correction in red)
That's correct!
Maybe I don't understand your reply correctly but in general the equations you gave represent straight lines which are placed in the plane you are looking for.$\displaystyle (x,y,z) = (a,b,c) + t(a-d, b-e, c-f)$
$\displaystyle (x,y,z) = (a,b,c) + t(3, 2, 1)$
I suppose you could combine them into:
$\displaystyle 2(x,y,z) = 2(a,b,c) + t(3 + a - d, 2 + b - e, 1 + c - f)$
$\displaystyle (x,y,z) = (a,b,c) + \frac{t(3 + a - d, 2 + b - e, 1 + c - f)}{2}$
EDIT:
But I'm not sure why you would use such an elaborate method.
$\displaystyle (x,y,z) = (a,b,c) + t_1 (3,2,1)$
$\displaystyle (x,y,z) = (d,e,f) + t_2 (3,2,1)$
You could use either one of those equations as well. If this plane passes through both points, and is parallel to a certain vector then you only need one of those points and the vector it's parallel to.