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Problem 68 Hard Difficulty

Which of the following four lines are parallel? Are any of them identical?

$ L_1 : x = 1 + 6t , y = 1 - 3t , z = 12t + 5 $
$ L_2 : x = 1 + 2t , y = t , z = 1 + 4t $
$ L_3 : 2x - 2 = 4 - 4y = z + 1 $
$ L_4 : r = \langle 3, 1, 5 \rangle + t \langle 4, 2, 8 \rangle $

Answer

$L_{1}$ and $L_{3}$ are parallel; $L_{2}$ and $L_{4}$ are parallel

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Video Transcript

the question they are asking which of the following four? Lines are parallel and are identical. So the Lines are given by the equations L one X equal to one place 60, Why Equal to one minus treaty? And they recorded 12 3 plus five. And I'll do is given by the equation X equal to one plus two, T Y equal to T Z is equal to one plus 40. And Elstree is given by the equation two x minus two equal to four minus four, Y is equal to that plus one is equal to T. C. and L four is given by the equation are equal to 315 Plus t. 4-8 in victor format. So in order to find the solution of the given question first we have to find the equation of each of the lines in regular forms. So and one line has a normal victor equal to six minus three, 12 and a point passing through it is equal to 115. Therefore the equation of the line is equal to six into X -1 -3, 2, Y -1 Plus 12 into that -5 is equal to zero. And after solving this equation though, Equation for L one Is equal to two, -Y plus or that is equal to 21. And similarly the Equation for L two can be written in the regular form as L two has the Normal vector is equal to 214 and the point that passes through this line is equal to 101 and therefore the equation of the Line, L two is equal to two into X -1 Plus one into Y -0 0 plus four into Z -1 equal to zero. After solving this equation, no final equation of L two is two, X plus y plus four. Zed is equal to six and the aggression for Elstree can written as X equal to t plus two by two, Y equal to four minus T by four and That is equal to T -1. Therefore the normal vector of this line is equal to 0.5 -1 x four and 1 as this is infraction form. So we can multiply this vector Bayous killer 1 24 so this is equal to two -1 and four. Therefore from the above equation we find up Point that passes through this line is equal to 1, -1. Therefore the equation of this line is equal to two into x minus one minus one into y minus one plus four in two, Z plus one is equal to zero. So the final equation for l C is equal to two, x minus y plus four, Z is equal to minus three. And The question for line L four is did I from the Normal victor that is equal to four 28. And the Point that passes through this line that is equal to 315. Therefore the equation of this line can be written as four into x minus three plus two into y minus one plus eight into zero minus five is equal to zero. And the final equation that can region for the line, L four is equal to two, x plus y plus or that is equal to 27. Mhm. So the equations of the line, L one, L 2, L three and L four. As we have already found out using the values of the normal vector and the coordinates from each of the equations that was given in the question to be written in the regular form. And so in order to find radio, these lines are parallel or identical. Mhm. Thank you. We have to check For the first pair of lines that is L one and L two. Yeah that is a condition is whether there are parallel or not. So in order to find their parallel or not to compare the coefficients of the X, y and z coordinates of these two lines. So this is equal to two by two is equal to minus one by one is equal to four by four. After simplifying these ratios, we get this equal to one not equal to -1, not equal to one. Therefore L one and L two R not parallel since they are not parallel, they are not identical. Also in the second condition we have to take for the next pair of lines that is that is L one and L three condition, he is whether there are parallel or not. So in order to find their parallel or not we have to compare the coefficients of each other coordinates of these two lines. So the issues are two x 2 is equal to -1 x -1, equal to four x 4. So after simplifying these ratios, we get It to be equal to one, That is equal to one equal to one. Therefore L one and else we are Palin hence we have to check whether these are identical or not to check it, we have to Put the value of x equal to y equal to zero in border equations of the line L one and else trees. So the first line L one here, the question would be that equal to 21 x four. And so therefore the point or The X, YZ coordinates would be 0, 0 and 21 x four. And therefore For the line three Putting the same conditions, that is equal to -3 x four and therefore your point is 00 and -3 x four. Therefore, since the coordinates are not equal, so L one and L three are not identical. Yeah. Yeah. And for the third case for the third line pair, that is L one and L four, we have to take the same conditions first, is there parallel or not? So comparing the coefficients, the coordinates of these two lines, we get the ratio is two by two equal to minus one by one equal to four by four. So after simplifying these ratios we get It will be equal to one is not equal to -1 Is not equal to one. Therefore L one and L four are not Berlin. Therefore these are not identical also to take next pair of lines that is L two and L three. The first condition to check whether these are valid or not. So in order to check we have to find the ratio between the coefficients of each other coordinates of these two lines. Therefore the Ratios are two x 2 is equal to one x -1 equal to four x 4. So after simplifying these ratios we get the value to be equal to one, not equal to minus one, not equal to one. Therefore L two and L three are not parallel and hence these are not identical. Also the next spirit of lying is and To an L four. Therefore taking the first condition that is whether these lines are parallel or not. So comparing the coefficients of the coordinates of these two lines, the ratio becomes two by two is equal to one by one equal to four by four, softer, simply paying these ratios speak get to be equal to one equal to one equal to one. Therefore L two and L four are parallel lines and hence we have to check the second condition in the question that is whether they're identical or not. So You put the value of x equal to y equal to zero To find the value of that and all the coordinates each of the lines. So for the line L- two that is equal to six by four. And Therefore the point for this is equal to 00 and six x 4. And the point for the line L four is XY equal to zero, so that is equal to 27 x four. So here the point is equal to 00, x four. Hence since .1 is not equal to point to, therefore L two and L four are not identical. Mhm I'm checking the last pair of lines that is L three and and four. So the yeah, first condition to check whether these lines are parallel or not. So the yeah condition to check this is we have to compare the coefficients of all the coordinates of these lines. So the issues are two by two Is equal to -1 x one equal to four x 4. Therefore, after simplifying the ratios we get to be equal to one is not equal to -1 is not equal to one. Therefore L three and L four, R not parallel. Hence these lines are also not identical. So from the above conclusions, we find the answer for this question to be L two and L four are parallel lines but not identical and L one and L three are also palette lines, I'm not identical, and this is the required answer to the given question.