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Problem 65 Hard Difficulty

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


$x=3 t y=1-t z=2-2 t$

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Video Transcript

In the question. They are asking to find parametric equations for the line through the .012. And the line is parallel to the plane Explains why place are equal to two and perpendicular to the line, X equal to one plus city by equal to 1 -10. That equal to duty. In order to find the parliamentary question on the line. It can written in the form as follows, that is X equal to except he is equal to egg zero less 80 Y T. Is equal to y zero plus Bt. And The T. is equal to the zero last city. Where this is the parametric equations. Off line asking the question met X zero Y 000 is given in the question to be equal to one, to be equal to 012. As the line passes through this point. So these coordinates are equal and the variable abc is the direction vector for the line that is V. Is the direction vector is equal to a B C. In order to find a parametric equation of the line. We had to find a direction vector of the line so to find Mhm. The direction vector, Yeah. We have been given to equations of line and plane that is first system but Berlin plane equation is given to the line. That is explains why place that equal to do This is equation one and second is given the or perpendicular. Okay, Mhm line equation to this line. No, as what X equal to one Plus T. Why equal to 1 -5. And they're equal to duty. So this entire set of parliamentary equation for the perpendicular line is named. Do so Using these two equations one and two, we have to find the direction vector B. Mhm. For the line for which we have to find a parametric equation. So in order to find the direction victor. Mhm. So using first using one equation which is given in the question to be Yeah. Yeah. Okay. So using the equation to that is a perpendicular line to this line. We can find the value in terms of a BNC. So the victor, Okay of the or a particular line, the uh Normal vector to the perpendicular line, let it be one. Victor is equal to the coefficients of the reliability in the parliamentary question of the perpendicular line. That is one comma minus one comma two. Therefore, if you find the dot product of the direction vector and the normal vector of this particular line. Okay. Mhm. Which is equal to a minus B plus to see equal to zero. From here we got equation which will be used to find the value of a BNC. What let it be named as equation A And from this first equation, that is the plane that is Palin to this line. The equation of the plane is expressed by play set equal to two. Therefore, yeah. Mhm Yeah. The normal victor. Mhm and cup to this plane is equal to 111. Therefore, since this plane is parallel two, the line which is contained in. Yeah, another clean. So if we compute the direction vector dot this normal vector of the parallel plane, then it will be zero as the direction vector of the parallel plane containing the line has the same direction vector as the line. And the normal victor is the perpendicular to the given plane. So the dot product will be called to zeros, Is there or terminal to each other? So the value of direction vector as we know, is A. B. C. And this. with the normal Victor 1. 1, 1 Is equal to zero. So here we get another question. In terms of abc that is A plus B plus C is equal to zero. We named the situation as situation be. Therefore equation A. And if we evaluate them That is a -7. Place to see Equal to zero. And a let's be let's see equal to zero. Then first we have to multiply the the second equation with two, we get to see to be to a And if you subtract each other terms to derive the answer and eliminate some variables, we get this. We see that the C term is canceled. And though value after computing this is minus minus a minus three, B is equal to zero. So this is equal to a plus three equal to zero or three. Big Wall. To -A. From here. If we calculate the value of C then A plus B plus C, we know equal to zero. Therefore putting the value of A. And B. To find the value of C. We get okay minus three B Plus B Plus C Equal to zero. Therefore this is equal to the A. C. Is equal to to be. Now if we assume a particular value of being Then if B equal to -1 then the value of the direction vector is equal to abc that is equal to three B me and to be so this is equal to three minus one and minus two. Therefore we find though parametric equation of the line asking the question, it will be equal to x. T. Is equal to x zero plus 80 YT is equal to Y0 Plus BT. The T is equal to The zero plus city. And as we know the value of X zero, Y zero and zero is given in the question. To be the point where the line is passing through is equal to 012. And therefore the parametric equations is X is equal to zero plus the direction vector. We know it is equal to 3 -1 -2. So it is equal to zero Plus 3 T. Why is equal to minus mhm. Mhm. Why is equal to one minus D? And similarly there is equal to 2 -2 T. So therefore this entire equation can be written in simplified form as x equal to treaty, why equal to one minus T and Z is equal to two minus duty. And hence this is the set of the parametric equation, which is asked in the question of the line that passes through a 20.12 and his palette to a plane. Having equation explains why plus and equal to two and is perpendicular to a line. Having a set of the equation that is X equal to one plus T. Y equal to one minus T and Z is equal to duty. Hence this is the required answer Okay?