5

ODOD IuJoQUESTION 5 [15 Marks]Given the lines /1 and I2W:I =1+3. Y = 2 - Gt , 2 =7t_3.= 6_ 5 _ = -812Show that /1 and 12 are skewed by showing that they do not inte...

Question

ODOD IuJoQUESTION 5 [15 Marks]Given the lines /1 and I2W:I =1+3. Y = 2 - Gt , 2 =7t_3.= 6_ 5 _ = -812Show that /1 and 12 are skewed by showing that they do not intersect and not parallel. marks] Find the equation of the plane containing the line /1 and parallel to the plane containing the line I2. Hence obtain the shortest distance between the lines /1 and /2. marks]Find the acute angle between the line /, and the plane 3r + Jy + 4 = 6_5 marks]

ODOD IuJo QUESTION 5 [15 Marks] Given the lines /1 and I2 W:I =1+3. Y = 2 - Gt , 2 =7t_3. = 6_ 5 _ = -8 12 Show that /1 and 12 are skewed by showing that they do not intersect and not parallel. marks] Find the equation of the plane containing the line /1 and parallel to the plane containing the line I2. Hence obtain the shortest distance between the lines /1 and /2. marks] Find the acute angle between the line /, and the plane 3r + Jy + 4 = 6_ 5 marks]



Answers

$51-56$ Determine whether the planes are parallel, perpendicular,or neither. If neither, find the angle between them. (Round to one decimal place.)
$$5 x+2 y+3 z=2, \quad y=4 x-6 z$$

In this problem. The normal vector of the given plane can be given by and one is equal to 14 minus three. And n two is given by minus 367 because then one is not is equal to and two so two planes are not balance. So I can write and one multiplication. And to is equal to one multiplication minus three plus poor multiplication six plus minus three multiplication seven. So I'm solving it. I get the value of minus three plus 20 for minus 21 which is equal to zero. So anyone an end to our perpendicular and their corresponding plans are also perpendicular. So here I can write anyone vector and and two factor are perpendicular and did goodness bonding planes. Uh, also perpendicular. So this is our answer for this question.

So with the planes were given, we can extrapolate our normal vectors. So it's going to be vector A two negative three negative one and vector B. Which is for negative six negative too. To determine if these planes orthogonal, We take their. and see that it equals 28. Since that's not zero. We see it's not orthogonal. However, we do see that this is parallel because this is a multiple. If we multiply this by two, we end up getting this so they're clearly parallel Which means the angle between them is 0°.. We could show this as well because Um the magnitude of a. is 14 magnitude of B is to route 14. So it's going to give you 28/28. The Inverse co sign of 28/28 Is obviously going to be 0°,, which is our fate of value.

What about the angle between the two factors? So normally will have? Of course. I'm the angle Tita. Nickel to the n one top. And to do you have any by the norm, the and gone attempts and numbed into the discussion were given the two plans. So the first plan is that you asked past three why Plus for that equal to five. The second plan is X plus six y, and then plus for is that equal to three. Now we can find a normal and one, and we go to the coefficient in front off the X y Z committed to monastery for and then we can find a number two. We go to the 1/6 and four and try to find a close eye on the angle. Between them, we go to the end one daughter and two. However, the number and one terms of phenomena and to and we get E co two here we have with two months 18 then plus 16. You're running by the squared off the four plus nine plus 16 time squared under one plus 36 plus nah, 16 on. Then it will simplify. We get the top beginning. Continue zero The writing by whatever here So study coaching a zero when the coaches Oh means that the English Cheetah you go to the Pyatt Thio, Niko to 90 degree. It means that the two plan the amber particular

Recall that that you planned that they have parallel should've gone gone that but allowed you to plan to even only if no, they're no more here will be parallel as well on even on the If the number and one can Britain, there is a multi bottle and chill in this question were given The two is sad ego to for why minus X and the second plan is three X minus trout. Why plus six? See you go to one. From the first question, we can run the minus X blissful y minus two C equal to zero. And from here we can write a normal and one they will go to the coefficient in front of the X y Z. Then we get equal to the minus one for minus two. And for the second plan, we're gonna end to it. We go to three minus trail on then six. Notice that we can write the into ICO June the, um, 13 times the and one. And from here we can conclude that that you planned their parallel


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