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Problem 43 Easy Difficulty

The figure shows a line $ L_1 $ in space and a second line $ L_2 $, which is the projection of $ L_1 $ onto the $ xy $-plane. (In other words, the points on $ L_2 $ are directly beneath, or above, the points on $ L_1 $.)
(a) Find the coordinates of the point $ P $ on the line $ L_1 $.
(b) Locate on the diagram the points $ A $, $ B $, and $ C $, where the line $ L_1 $ intersects the $ xy $-plane, the $ yz $-plane, and the $ xz $-plane, respectively.


(a) $(2,1,4)$
(b) $N=L_{2} \cap O y$
$B \in L_{1}$ such that $N B \perp x y$ -plane


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Video Transcript

Hello and welcome. We're looking at, uh, problem or chapter eight. Section one, this is Problem. 35 were asked two questions to parts of question. First, let's find the coordinates of the point P on the line. L one. So here's our line. L one we wanna guess we kind of estimate, or we can use the line l two to our advantage. So we know that, um, all the points on L to r directly below straight line down below all the points on l one. So if we can't if we draw a vertical line down, that's pretty good right there. This point looks we know this line l two is completely on the X y plane. So this point right here looks to be at on X value of two. Anna. Why? Value of one kind of 2/3 of the way. Done that. Now we just need to make a guest for the Z value s 021 You could make an argument for ah Z value of three, but I think we have a Z value of four. The value of four. And that's because this is coming. This is coming forward. So if we kind of extend, you can see as equals zero to get to like, X equals two. It's coming down and forward. So if we come down and forward from three, that's too low. If we come down and forward from four just trying to think spatially in three dimensions that seems to be is the value of force of heart, eh? We have according to 14 All right, um, so for part B, See, we want to locate on the diagram points A, B and C where l one are Red line here intersects with the various x Y planes Y Z plane at X Z plane. I'll do that in blue here. So, er the x Z plane. Well, let's start with a is where it intersects the X Y plane. Um, so the line l one intersex the x y plane, right? Uh, would be where these two lines now, that's because we know all of these lines are, um, sorry. We know everything on L two is directly blow everything on l one. So the point where they intersect is actually the point where they intersect. That's where that's where the Red line crosses through the X Y plane. So this right here would be a B is where crosses through the Y Z plane where it crosses through the Y Z plane would be directly below where l to process the y Z play across is the y axis. Um, so if we go directly below, that's down towards the bottom of our screen here. That would be be, uh, because where l two crosses that Y Z plane. Um l one has to cross it at the same place because it's directly below. Finally, point C is where we're crossing the Z plane. This wall here. Well, so we want to look at where l two crosses the X Z plane, and we know if we go directly above that's the point where l one crosses the X Z plane. Right? Wait.