Determine whether the lines $ L_1 $ and $ L_2 $ are parallel, skew, or intersecting. If they intersect, find the point of intersection.
$ L_1 : x = 3 + 2t , y = 4 - t , z = 1 + 3t $
$ L_2 : x = 1 + 4s , y = 3 - 2s , z = 4 + 5s $
according to the question we have to determine whether the lions, L one and L two R parallels Q. Or intersecting If the intersect, find the point of intersection, The given lines are L one has the direction coordinates as X equal to tripolis duty by equal to four minus T. Is equal to one plus three. T. And L two is equal to X is equal to one plus forest. Why equal to three minus tourists. And that equal to four plus five is So according to the criteria they're asking to search. That is whether these lines are parallel or not. So these lines has coefficients off the unknowns. That is given in each of the coordinates. If we find the ratio of each of the coordinates for each of the lines then if they are all equal then only these two lines are parallel to each other so gradually one is checking for L one and L two are Caroline two. Each other are not. Yeah. So the coefficients of each of the coordinates and the rest of the two lines are two by four is equal to minus one x -2 equal to three x 5. These numbers are taken from the coordinates of each of the lines. That is so coordinate of the X coordinate and coefficient of the first lane and this is the coefficient of the X. Coordinate of the 2nd Line. Yeah. In this way all the coordinates are put in ratio form to find whether these lines are in same direction to each other or not. Okay, so if we calculate this then it will Simplify as one x 2 equal to one x 2 equal to three x 5. Since she verified is not equal to one x 2. Therefore L one and L two R not parallel to each other. Okay now we'll take the second criteria. That is my book whether L one and L two are intersecting or not. Mhm. Okay. Yeah. Mhm. So in order to check this we have to equip all the coordinates of borderlines that is you have to equipped X one equal to X two. Why one equal to Y two and okay that one equal to that too. Therefore if we substitute any particular value in other equations and if it satisfies the equation then we can say that these to answer intersecting to each other. Otherwise it will not intersect. So the equations are three Plus to be equal to one plus for us or minus T is equal to 3- to us and Yeah. Okay one plus three D. Is equal to four plus virus. So from equation one we can find the value of T. That is to t is equal to one minus three plus four. S that is two days equal to For S -2 or T is equal to The West -1. Okay. Yeah. Yeah. Yeah I'm putting this value of t. Any question. Do we get? Okay. Mhm. Yeah. Yeah. Yeah. What? Yeah. Immigration. Do we know that four minus T Is equal to 3- to us. Therefore if you substitute the value of teeth from equation for that is physical too. Mhm two is -1. In this equation. We get that four minus two. Waste minus one is equal to three minus to us. Yeah. Therefore 4 -2 s Plus one is equal to 3 -2. Is or We can check that -2 s. is comin from both sides so they will cancel each other and the equation becomes five equal to three or To equal to zero which is not true. Therefore L. one and L. two. Oh not intersecting also. Okay. Yeah. Yeah The 3rd criteria is to check. Yeah. Mhm. If L one and L two R Schiewe lines mm Therefore as L one and L two are not parallel or not intersecting yeah to each other that being said. Therefore L one and L two lines. I skew lines listen and since L one and L two lines are not intersecting so we cannot find the intersecting points off L. one and L two. Mhm. As they are skew lines. So of course this is the required answer