- #1
crazycat23
- 2
- 0
Homework Statement
The point A(2,4,5) is reflected in the line with equation r = (0,0,1) + s(4,2,1) , sER, to give the point A'. Determine the co-ordinates of A'.
Homework Equations
1. The equation of the line (parametric equations)
2. A (2,4,5)
Dot Product, and distance from a point to a line
The Attempt at a Solution
I first began by finding the distance between r and A, equating it to r and A'. With A' being (a,b,c), and substituting this into the parametric equations of the line, i got the following:
824 = 5a^2 + 17b^2 + 20c^2 - 4bc - 8ac - 16ab
I'm unsure where to proceed from here, or even if this equation is correct.
---------------------------------------------------------------------------
Homework Statement
For the lines L1 and L2, determine the co-ordinates of points on these lines that produce the minimal distance between L1 and L2.
Homework Equations
L1: r = (1,-2, 5) + s(0,1,-1), sER
L2: r = (1,-1,-2) + t(1,0,-1), tER
P1 (lies on L1) = (a,b,c)
P2 (lies on L2) = (d,e,f)
The Attempt at a Solution
I know that the distance from each point to it's opposite line is equal, and I also broke up each line equation into parameters and substituted its respective point. I arrived at this answer however, and looking back, I can't figure out my thought process as to how I got to it, so I wouldn't trust it. lol
distance = (2t^2 +2s^2 +14t - 14s - 2st + 49)^1/2
-----------------------------------------------------------------------------------
Homework Statement
Point A(1,0,4) is reflected in the plane with equation x - y + z - 1 = 0.
Determine the co-ordinates of the image point.
Homework Equations
The plane equation
A(1,0,4) and it's reflection B(a,b,c)
The Attempt at a Solution
Again, distances are equal. I have two distrance from a point to a plane formulas equated to each other. I end up with this:
4 = l a - b + c - 1 l
I'm not exactly sure where to proceed from here.
-----------------------------------------------------------------------------
Homework Statement
A perpendicular line is drawn from point X(3,2,-5) to the plane 4x - 5y + z - 9 = 0 and meets the plane at point M. Determine the co-ordinates of point M.
Homework Equations
The points X, M and the plane equation
The Attempt at a Solution
Since the distance from a point to a plane is perpendicular, I found the distance from X to the plane as 12/(42^1/2).
I converted the plane equation into a vector equation, and then broke that into parametric equations. Here were mine: x = s + m, y = -s, z = 9 - 9s + 4m, with s and m being the parameters.
Using the distance between two points formula, and equating that to my distance, substituting the parameters, I receive this equation:
7a^2 + 7b^2 + 7c^2 - 42a - 28b + 70c + 242 = 0
Again, I don't know where to proceed
----------------------------------------------
Sorry I have so many questions. I hope you can help me, especially with my problem with finding equations and then getting stumped!