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Problem 66 Hard Difficulty

Find parametric equations for the line through the point $ (0, 1, 2) $ that is perpendicular to the line $ x = 1 + t , y = 1 - t , z = 2t $ and intersects this line.

Answer

$x=1.5 t, \quad y=1-0.5 t, \quad z=2-t$

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Video Transcript

the question they're asking to find the parliamentary equations for the line That passes through the .012 and this line is perpendicular to the line X equal to one, testify equal to one minus T and they're equal to 2. 10 intersects this line. So in order to find the parliamentary equation of the line first, we had to find the form of the equation. The line will be. So the parametric equation of the line is written as X. Of T is equal to X zero plus 80 Y T is equal to y zero plus Bt and Z T is equal to the zero plus city where egg zero, Y zero and 00 is still coordinate that passes through the line. And that has given me the question 01, two. And so in order to find the parametric equation, we have to find out values of the direction vector that is equal to abc. So in order to find the value of abc first, we have to find the direction vector of the line that is perpendicular to this line. That is the line is given by the equation. Yeah, X equal to one plus T. That y equal to one minus T and they're equal to duty. So the direction vector for this line is Be one Victor is equal to 1 -1 and two. Therefore, if we find the dot product of B and we won them as a perpendicular. So this is equal to zero. Therefore abc 0.1 minus 12 is equal to zero. Therefore the value in terms of abc. And written in the form of the equation A plus a minus B Plus two, C is equal to zero. And this is the equation one. In terms of ACC for calculating the values of the direction vector of the line and another. And another point is also given the question that this line, this perpendicular line also intersects the line asking the question so far this criteria we can evaluate the coordinates to be equal. So X one equal to X two, Y one equal to y two and then one equal to that too. Therefore Here X one is equal do 80. This is equal to X two, That is one plus T. Why one is one plus bt that is equal to one minus T and That one is two plus city, That is equal to two T. The two. And after equating and evaluating first, the value from the first equation Yeah, can be written as T is equal to one by a -1 from equation one and this is supposed equation A. And next from aggression do we can find another value that is T into a B plus one equal to zero or Be equal to -1. Okay, as you mean. Okay, b to be a constant value. Therefore this is the value found from equation to let it be be. And From equation three we can right that yes two plus city is equal to two T. Therefore from here we find that T. Is equal to -2 divided by C -2. Let us suppose the situation be turned as C. Therefore from mhm A and C equation we can equate them equal as both are equal to T. So one by a minus one is equal to minus two by c minus two. After computing this value find equation which is equal to C Plus two is equal to fall. Yeah and hence this equation we can rearrange in the form, Find the value of a. That is equal to four -C by no. So let's suppose this question be equation D. So Mhm. Yeah from the previous equation that is yeah That this equation one yes a minus B plus to c equal to zero. So a minus B plus to c equal to zero. And another equation that we just got from here is simplest week. Well before so see Plus two ways equal to four. So in order to find the value We multiply the first equation by two and so to a minus to B plus yeah Foresees equal to zero. So after subtracting the below equation from the above equation we get the value of a cancels out and so minus to b minus plus three. C is equal to minus four. So as we know from the equation be that Be equal to -1 Since be equal to -1. Therefore we can find the value of. See from this equation Therefore 2-plus 3 is equal to -4. Therefore from here C is equal to minus two. So we got the value see value and now from the equation B. That is equal to four -C by two. She We can find the value of a tons of C which we got the value is -2 so is equal to four plus two by two. That is equal to three. So we got the direction vector of the line. So be vector is equal to a B. C is equal to 3 -1 -2. So since better the notes direction off. Mhm Any object? So mhm. The skill er one duty can be okay changed according to Mhm. Yeah. Mhm requirements that is multiplied bye up on stand value are scalar value. So if we divide this victim quantity by two then mhm This can be written us 1.5 minus 0.5 and minus one. Therefore the parametric equation. Yeah. Yeah. Mhm. The parametric equation of the line can be written us XD is equal to mhm Egg zero plus 80 White is equal to Y0 plus BT And the T. is equal to the zero plus city. Therefore the final equation. Mhm For the line that this parametric informed is equal to X equal to for the point that is yeah contained in the line given in the question as 012 So X is equal to zero plus 1.5 G. Why equal to one plus -150? and that is equal to two Plus -1 T. Therefore, yeah. Yeah. The final set top equation, which is the parliamentary equation of the line asked in the question is X equal to 1.5 G. Why equal to 1 -150? And that is equal to two -T. So this is now answer of the given question yeah.