Analytic Geometry Problem - find a line in space meeting some conditions

In summary, Antonio is seeking help with a geometry problem involving finding the equations of lines lying on a plane and intersecting a given line at a specific angle. He initially attempts to solve the problem by considering the intersection point and a generic point on the z-axis, but later realizes a fault in his approach and makes corrections. He is seeking suggestions and corrections from the forum members.
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Hi all! I've appreciated (and lurked :-) this forum for a while. While waiting to find a question I'm able to answer, it's time for my first post (hopefully the first of many).

Here's the problem: I'm trying to teach myself euclidean geometry and a found a tough (for me) question to answer in the textbook I use. As you can read below, I attempted to find a solution but I'm not sure it fits. So I'm asking you help me find out any faults in my approach to the problem. Sorry for the bad english, but I'm just trying to translate:

Find the equations of the lines lying on the plane [tex]\pi: x-y+z=0[/tex] intersecting the line [tex]r:\left\{\stackrel{x=-z+2;}{y=-2z+2}[/tex] and making a [tex]\frac{\pi}{4}[/tex] angle with the z-axis



The attempt at a solution
I tried to solve the problem by noting that the line(s) we are looking for, by lying on the plane [tex]\pi[/tex] and intersecting the line r must share with them one point, given by the intersection of the plane and of the line r itself.

So I found out it had to pass throug P(2,2,0).

Then i tought it would be useful to consider the vector [tex]\bar{PZ}[/tex] with Z being the generic point of the z-axis given by (0,0,k).
The direction of the z-axis is given by [tex]\bar{z}(0,0,1)[/tex].

So since given two vectors u and v we have that cos uv = [tex]\frac{|uv|}{|u||v|}[/tex], i tried to do [tex]\frac{\sqrt{2}}{2}=\frac{k}{\sqrt{8 + k^{2}}}[/tex]

Substituting I found out that k=[tex]\pm2\sqrt{2}[/tex]. That's because cos (PZ z) had to be [tex]\frac{\sqrt{2}}{2}[/tex].

Substituting again, the line(s) I'm looking for should pass from P(2,2,0) and K(0,0,[tex]\pm2\sqrt{2}[/tex]). This gives me two lines, as I had to expect.

And that's all. I'm just waiting for your suggestions/correction (expecially corrections :-).

Thanks in advance,
Antonio

PS: sorry for bad English and formatting... I have to get used to both :-)
 
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  • #2
Hi again...
I just found a fault in my attempt to solve the problem... I didn't consider that (0,0,[tex]2\sqrt{2}[/tex]) actually doesn lye on the plane [tex]\pi[/tex]! So i tought to impose that the x and y coordinateof of the point would be such that [tex]x - y + z = 0[/tex], with z being of course [tex]2\sqrt{2}[/tex].
Given this, everything else seems to be ok.

Again, thanks in advance for a reply,
Antonio
 

Related to Analytic Geometry Problem - find a line in space meeting some conditions

What is analytic geometry?

Analytic geometry is a branch of mathematics that combines algebra and geometry to study the properties and relationships of geometric figures using coordinates. It involves representing geometric objects and their properties using algebraic equations and formulas.

What are the main components of an analytic geometry problem?

The main components of an analytic geometry problem are a set of coordinates, equations or inequalities, and geometric figures such as points, lines, and curves. These components help to define the problem and guide the solution process.

How do I find a line in space meeting certain conditions?

The first step in finding a line in space meeting certain conditions is to identify the given conditions and determine their mathematical representation (equations or inequalities). Then, use algebraic techniques such as substitution or elimination to solve for the coordinates of the line. Finally, plot the points and connect them to form the desired line.

What are some common conditions given in analytic geometry problems?

Some common conditions given in analytic geometry problems include the slope of a line, the distance between two points, the angle between two lines, and the intersection of two lines or curves. These conditions help to define the problem and provide a starting point for finding a solution.

Are there any special techniques or formulas for solving analytic geometry problems?

Yes, there are various techniques and formulas that can be used to solve analytic geometry problems. These include the distance formula, the midpoint formula, slope-intercept form, and point-slope form. It is important to understand these techniques and when to apply them in order to solve problems efficiently.

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