Need help with four questions

[accordingplum]

"Question 1: A point is mirrored in the line 4x + 3y = 0 and the position of the new point is (-5, 3). What is the position of the original point?"

I tried to solve this problem and the answer I got was (-945/649, 3650/649) but it feels wrong. What I did to get this answer was by first calculating the transforming matrix of the line and then I calculated the inverse matrix and multiplied it with the point (-5, 3). Have I solved the problem and if I didn't, what did I do wrong?

"Question 2: A light beam that originates from the point (3, -2, -1) is mirrored on the plane x - 2y -2z = 0. The mirrored light beam goes through the point (4, -1, -6). In what point did the light beam hit the plane?"

I have tried to solve this problem but I'm stuck. First, I created two vectors: v and u. v = (3-x, -2-y, -1-z) and u = (4-x, -1-y, -6-z) where (x, y, z) is the point on the plane where the light beam hits the plane. Then, I multiplied v and u with the plane's normal (n). For each multiplication, I got two equations: n ⋅ v = |n| ⋅ |v| ⋅ cos θ = x₁x₂ + y₁y₂ + z₁z₂ and
n ⋅ u = |n| ⋅ |u| ⋅ cos θ = x₁x₂ + y₁y₂ + z₁z₂. What I realized was that the angle θ is the same for both equations so I isolated cos θ in one of the equations and replaced the other cos θ with what I had equated cos θ with in the other equation. And here's where I'm stuck. How can I approach this problem to solve it?


"Question 3: Two beams on a construction site can be described as
s1 = {x = 1 - 2t, y = 2 + 6t, z = 4 - 8t 0 ≤ t ≤ 5
and
s2 = {x = 12 + 2t, y = 4 + t, z = -8 + t -10 ≤ t ≤ 10

These two beams should be connected by a third beam that is perpendicular to both s1 and s2. In what points on s1 and s2 should the third beam be attached?"

I realize that I have to find a normal that is perpendicular to both lines but I don't know how. Any help is appreciated.

"Question 4: u, v and w are three vectors in three-dimensional space. |u| = 5, |v| = 2 and |w| = 1. The angle between u and w is π/3. What should the angle between u and v be so that |u + 2v + w| = |u + v + 2w|?"

I actually haven't started on this one but I can already tell that it's a very difficult problem. Any tips on how to solve this would be appreciated!

 

[MarkFL]

Hello, and welcome to FMH!

For the first problem, I would look at the slope of the given line, which is:

[MATH]m=-\frac{4}{3}[/MATH]
And so the point we want will lie along the line perpendicular to the given line and through the given point:

[MATH]y=\frac{3}{4}(x+5)+3=\frac{3}{4}x+\frac{27}{4}[/MATH]
Now, we find the two lines intersect at:

[MATH](x,y)=\left(-\frac{81}{25},\frac{108}{25}\right)[/MATH]
The square of distance \(d\) between the two points (the given point and the point of intersection) is:

[MATH]d^2=\left(-5+\frac{81}{25}\right)^2+\left(3-\frac{108}{25}\right)=\left(\frac{11}{5}\right)^2[/MATH]
To find the original point, we need the other point on the perpendicular line that same distance from the intersection:

[MATH]\left(\frac{11}{5}\right)^2=\left(x+\frac{81}{25}\right)^2+\left(\frac{3}{4}x+\frac{27}{4}-\frac{108}{25}\right)^2[/MATH]
This implies:

[MATH]x\in\left\{-5,-\frac{37}{25}\right\}[/MATH]
And thus:

[MATH](x,y)=\left(-\frac{37}{25},\frac{141}{25}\right)[/MATH]

 

[Dr.Peterson]

"Question 1: A point is mirrored in the line 4x + 3y = 0 and the position of the new point is (-5, 3). What is the position of the original point?"

I tried to solve this problem and the answer I got was (-945/649, 3650/649) but it feels wrong. What I did to get this answer was by first calculating the transforming matrix of the line and then I calculated the inverse matrix and multiplied it with the point (-5, 3). Have I solved the problem and if I didn't, what did I do wrong?

Please show the specifics of your work, so we can see whether your method is right, and where you might have made a mistake. In particular, what are the matrices you used, and how did you obtain them?