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Problem 3(25 points) Detemmine whether the lines3242 "-15 Li : 2-14andLz : 2t46 y+l4 2+17Intersect are skew, Or are parallel It they Intersect; determine the p...

Question

Problem 3(25 points) Detemmine whether the lines3242 "-15 Li : 2-14andLz : 2t46 y+l4 2+17Intersect are skew, Or are parallel It they Intersect; determine the point of Intersection; not leave the remaining answer blanks empty: The linesPoint of intersection:

Problem 3 (25 points) Detemmine whether the lines 3242 "-15 Li : 2-14 and Lz : 2t46 y+l4 2+17 Intersect are skew, Or are parallel It they Intersect; determine the point of Intersection; not leave the remaining answer blanks empty: The lines Point of intersection:



Answers

$19-22$ Determine whether the lines $L_{1}$ and $L_{2}$ are parallel, skew, or intersecting. If they intersect, find the point of intersection.

$L_{1} : \frac{x-2}{1}=\frac{y-3}{-2}=\frac{z-1}{-3}$
$L_{2} : \frac{x-3}{1}=\frac{y+4}{3}=\frac{z-2}{-7}$

Based on what we're given. The first thing we want to do is find the directional derivatives. So we look at the denominator and we see that our first factor is going to be 1, -3. And then we look at the denominator of the second equation and we see that our second factor will be 137 negative side. That is so we want to check to see if the vectors are parallel. Um and that's by using the cross product or by proportion proportion a little bit easier. We see the one times 1 is 1/2 to get to two times one is not three. So based on this, we see that they are not parallel, so not parallel. Then we want to see if the vectors are intersecting. So to do that we create parameters so we will have X equaling T plus two, why were equal negative two, T plus three And Z is equal -3. This should be a negative to negative two and then this is going to be negative three key plus one and that's based on both the denominator and numerator. Then we also get for the second group X equals S plus three Y equals three S -4 and Z equals negative seven S. Plus two. So substituting the S. And T. What? We end up getting our uh three equations T. Plus two equals S. Plus three. Then we get negative two. T plus three equals three. S minus four negative three T. Plus one equals seven S plus two. So our final answer is gonna be four negative one negative five for the intersection.

In this question were given the question that utilized So we have the like. One even is the thanks Pickle 23 plus two t why you go to the form honesty, the ego, You know, one plus three d and we have the second question on the line to But we have the X echoed Jew one plus for us. Why equal to three months to ask and Zico Junior Far plus five s And now let's try to serve the system in question here. So we're going to put the X equal to X. Why ego to why Onda the ego to Z Yeah, so it means that we have the system it questions now three plus two t we go to one bless for I ask for months Stay If we go to the three months to ask one plus three t equal to four plus five s Now, from the first question we can get the we can go from the second question here. So, thio every ego June the, uh, four months three. Could you one plus Thio ask. So just in this thing here, we can look into the first under say that question now to get from here. From the first equation began the three plus Tutino equal to the blessed to place. For this they go to one glass for us. So that's when we see that we we can concerned that for asking for us So doesn't get the five. You go to one citizen Impossible. Here. Therefore we can conclude that that you like one, um, like you Ah, scale because we have no solution here.

Question will even the 1st and 2nd question on the line. So when the and one it will be under form X under one they go to while I minus one over minus one Nico Jizzy managed to over three notice that we can tell us into the promise. The question Mexico 23 Why equal to one dynasty and Z equal to the two plus treaty? And now we have the equation to it will be X managed to over to we go to one ministry our minus two equal to see over seven again we continue agenda X equal to the two plus two and use the s Now why equal to the three months to s and the ICO to the seven asked. So we need to find the intersections on the July's means that we need to put the X equal to X So means GE go to the to bliss to ask why it could you know why? So we have the one Manistee included the three months to ask and see ico choosy. So when the two plus three t Nico to the seven s Yeah, I noticed that we can use ISTEA We're looking to the second question together. One months, two months to ask. You go to the three months to ask. So from here we see we can consider that you asking that you ask. So this means that we have a man is one of the co 23 So it's impossible The phone weekend that utilized and one and don't you ah, skill.

So here we see that we want to find the directional derivatives first and that's gonna be based on the denominator. So it's gonna be Warren -1, 3. And then our second vector is going to be To -27. Then we want to find if the vectors are parallel. So we see that um To get to here we multiplied by two. To get your we multiply by two. But to get to hear if we multiply by two at equal six. So because they're not a multiple of each other, that means that they are not parallel, so parallel is not. And then we wanted to determine if they're intersecting. So to do that we need to create parameters. So we end up getting X equals T. Y equals negative T plus one and Z equals three T plus two. And this is based on the numerator and the denominator of the first group. And then the second group gives us to S plus two. Second group gives us negative to S. Plus three. And then the 3rd group just gives us seven s. Then using these parameters, we want to check if the vectors are intersecting. So we have T equals two. S. Plus two negative T. Plus one equals -2 X. Plus three, and three T. Plus two equals seven S. So based on this we solve for S. And T. Through elimination. Um And once we do that we find that t minus t minus. Did you ask? Plus two Ass equals two plus two. These cancel these cancel, that means zero equals four, which is not true. That means that they are not. They do not intersect so no intersection, which means that these para lines are skew.


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