Determine the vector equation of a plane that contains the line with symmetric equation (x+3)/-2=(y+1)/5=(z+2)/4 and the point P(1,3,0).
For this question to find the direction vector you would use subtraction with the points (-3,-1,-2) and (1,3,0). Why would it be (1,3,0)-(-3,-1,-2) instead of (-3,-1,-2)-(1,3,0)? is it because you need to subtract the smaller number from the larger one?
A point on the line is (-3,-1,-2) and the direction of the line is [-2,5,4]
To find the equation of a plane I need another direction vector,
using the point (-3,-1,-2) and the other point P(1,3,0) , I get [-4, -4, -2] or we could simply use
[2,2,1]
since you want the vector equation, we can now just state that using parameters s and t
r = (1,3,0) + s[-2,5,4] + t[2,2,1]
of course you know that this equation is not unique.
your question about the direction of the subtraction is a mute one, of course it makes no difference, we could have used the vector
(4,4,2) or (-4,-4,-2) or (-2,-2,-1) or (8000,8000,1000)
Well, the reason we subtract (1,3,0) from (-3,-1,-2) instead of the other way around is not about finding the smaller and larger numbers. It actually has to do with finding the direction vector of the line.
When we subtract (1,3,0) from (-3,-1,-2), we are essentially finding the displacement vector from (1,3,0) to (-3,-1,-2). This gives us the direction in which the line is pointing. We can think of it as "starting from (1,3,0), how do we get to (-3,-1,-2)?"
On the other hand, if we subtracted (-3,-1,-2) from (1,3,0), we would be finding the displacement vector from (-3,-1,-2) to (1,3,0), which would give us the opposite direction of the line. It's like asking "starting from (-3,-1,-2), how do we get to (1,3,0)?"
So, to find the correct direction vector for the line, we subtract (1,3,0) from (-3,-1,-2). Hope that clarifies it!
No, the choice of subtracting (1,3,0) from (-3,-1,-2) instead of the other way around has nothing to do with the relative sizes of the numbers.
To find the direction vector of a line, one typically subtracts the initial point of the line from any other point on the line. In this case, the given symmetric equations represent a line passing through the point (-3,-1,-2) and parallel to the direction vector (2,5,4).
To find the direction vector of the line, we can choose any other point on the line. In this case, we can choose point P(1,3,0). Therefore, the direction vector will be (1,3,0) - (-3,-1,-2), which simplifies to (1+3,3+1,0+2) = (4,4,2).
To determine the direction vector of a line, you subtract the coordinates of one point on the line from another point on the line. In this case, the line is defined by the symmetric equation (x+3)/-2=(y+1)/5=(z+2)/4, and the given point on the line is P(1,3,0).
To find the direction vector, you subtract the coordinates of P(1,3,0) from another point on the line. Let's denote another point on the line as Q. The direction vector would be Q - P, where Q can be any point on the line.
Regarding your question about the order of subtraction, it does not matter in this case. Whether you subtract (-3,-1,-2) from (1,3,0) or vice versa, you will ultimately get the same direction vector. The direction vector only gives information about the direction of the line, not the length or magnitude.
So, you can choose any point on the line, subtract the coordinates of P(1,3,0) from that point, and the resulting vector will be the direction vector of the line.