Finding Points of Intersection and Proving Parallelism with Vectors

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SUMMARY

This discussion focuses on finding the intersection of the line defined by the parametric equations x = 3 + 2t, y = 7 + 8t, z = -2 + t with the coordinate plane, specifically the XY plane. The intersection point is determined by setting z = 0, which yields the intersection at (9, 23, 0) when t = 2. Additionally, the discussion covers proving that the line segment joining the midpoints of two sides of a triangle is parallel to and half the length of the third side using vector addition, establishing the relationship through the equation (1/2)(a + b) = (1/2)c.

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  • Understanding of parametric equations in three-dimensional space
  • Knowledge of vector addition and properties of vectors
  • Familiarity with coordinate planes, specifically the XY plane
  • Basic concepts of triangle geometry and midpoints
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  • Study vector operations and their applications in geometry
  • Learn about parametric equations and their intersections with planes
  • Explore the properties of triangles and midpoints in Euclidean space
  • Investigate the concept of vector parallelism and its proofs
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StonedPanda
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How would I find the points of intersection of the line x= 3+2t , y= 7+8t , z=-2+t , that is, l(t) = (3+2t,7+8t,-2+t) with the coordinate plane?

Also, how would I prove using vectors that the line segment joining the mdpoints of two sides of a triangle is parallel to and has half the length of the third side? I'm going to try this one some more before I look for the answers, but please give me some guidance for both of them!
 
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Figured it out!

If z=0, then you have the XY plane. So, find what T value makes z zero (2 in this case).
Then, plug in 2 for t in x and y (you get 9 and 23 respectively). Thus, the point it intersects the XY plane is (9,23,0).

That feels good!
 
Make it so that a points upwards and right, make b point down and right, and make c = a + b. You know the line segment joining the midpoints is:
(1/2)a + (1/2)b
= (1/2)(a + b)
= (1/2)(c)... Q.E.D.

For the first one, I'm not sure what you're referring to by "the co-ordinate plane," but you seem to have gotten that one, so you're fine.
 
Scuse me akg! It should have been Coordinate Planes =p mb

But, wouldn't c = a - b if a points upright and b points downright?
 
Not if you define c as a + b!

cookiemonster
 
StonedPanda said:
Scuse me akg! It should have been Coordinate Planes =p mb

But, wouldn't c = a - b if a points upright and b points downright?
Nope, it would be as I wrote it. Think about it: start at the "starting point" of a, and go in the direction that it goes in. Now, since you're adding b, you then proceed in the direction that b goes in, which is down. In this process you will have started at the "start" of c and ended at the "end."
 
Ok, I get it. If a points downleft and b points downright, then c is b-a . Is this correct sir?

Btw, thanks for your help akg and cook!
 
Yes, that's correct.
 
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Vectors are incredibly powerful. I've taken 4 classes of multivarient calc so far, and my knowledge of math increased exponentially!
 

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