Finding Points of Intersection and Proving Parallelism with Vectors

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In summary, the first question asks how to find the points of intersection of a line in three dimensions with the coordinate plane. The second question asks for guidance on proving that the line segment joining the midpoints of two sides of a triangle is parallel to and has half the length of the third side using vectors. The conversation also includes a discussion on the correct definition of c when a and b point in different directions.
  • #1
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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  • #2
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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!
 
  • #4
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.
 
  • #5
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?
 
  • #6
Not if you define c as a + b!

cookiemonster
 
  • #7
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."
 
  • #8
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!
 
  • #9
Yes, that's correct.
 
  • #10
Vectors are incredibly powerful. I've taken 4 classes of multivarient calc so far, and my knowledge of math increased exponentially!
 

What is a line intersecting a plane?

A line intersecting a plane is when a straight line passes through a two-dimensional plane at a specific point, creating an intersection.

How do you find the point of intersection between a line and a plane?

The point of intersection between a line and a plane can be found by solving the equations of the line and the plane simultaneously. This can be done by substituting one equation into the other and solving for the variables.

What is the relationship between a line and a plane in three-dimensional space?

In three-dimensional space, a line and a plane can either be parallel, intersecting, or skew (neither parallel nor intersecting). This relationship is determined by the slopes of the line and the plane.

Can a line intersect a plane at more than one point?

Yes, a line can intersect a plane at more than one point if the line lies within the plane or if the line is perpendicular to the plane.

What is the significance of a line intersecting a plane in real-world applications?

The concept of a line intersecting a plane is important in fields such as geometry, engineering, and physics. In geometry, it is used to determine the relationship between two geometric figures. In engineering, it is used to calculate the point of intersection between structures. In physics, it is used to study the motion of objects in space.

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