Line of Intersection of Two Planes

In this video we look at a common exercise where we are asked to find the line of intersection of two planes in space. As long as the two planes are not parallel to each other, there will be a line of intersection. In this video we go through the algebra for how to find this line of intersection.
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[@Patrick_Knowlton]

I'm using this to find out how many each of three different crop seeds I should buy in Stardew Valley...... it feels so strange using math like this in a casual scenario lol

[@Patrick_Knowlton]

ok so here's how it went down: Variables: x=number of eggplant seeds y=number of bok choy seeds z=number of yam seeds Formulas: x+y+z=104 (number of farm tiles I had plowed) 20x+50y+60z=4000 (each number is the price of one seed, and my budget was $4000) My process: 1. Solved for x x+y+z=104 -> x=104-y-z 20x+50y+60z=4000 -> x=200-2.5y-3z 2. Used both of these to solve for z 104-y-z=200-2.5y-3z -> 2z=-1.5y+96 -> z=-0.75y+48 3. Plugged in z to get x x=104-y-z -> x=104-y-(-0.75y+48) -> x=-0.25y+56 4. And obviously y=y 5. I chose a random number for y, to begin with (20). y=20 x=-0.25y+56 -> x=-0.25(20)+56 -> x=-5+56 -> x=51 z=-0.75y+48 -> z=-0.75(20)+48 -> z=-15+48 -> z=33 So in the end, I bought my 104 seeds; 51 eggplants, 20 bok choy, and 33 yams. It cost me exactly my $4000 budget. It was incredibly cathartic and satisfying to reach my answer!

[@QuickGrowthBites]

@@Patrick_Knowlton I love this so much hahaha

[@athena4838]

Omg wow...glad to know this has practical use😊

[@azyle2104]

@@Patrick_Knowlton oh wow , nice

[@Yeahagreed]

​@@Patrick_Knowlton did u get it right?

[@WaitinForAMate]

That is such a great technique to finding intersection lines between two planes. Thank you so much for your clear and easy to follow lesson :)

[@kshitiz6376]

Nice technique Prof. I was trying to use the cross product of normals to find the direction ratios of the line of intersection. Although,I got stuck how to proceed from there,your little lecture helped me. Thanks,from India.

[@TabooTalz]

Thank you so much, I've tried two other methods to learn this (including the one my text book and teacher recommend) and I've never understood this until now. It's worth 5 marks usually so I really needed this

[@jamielambrecht9419]

Best video on this topic.. Others were driving me crazy taking forever to explain it in convoluted ways. It could be found just by doing RREF on augmented matrix to find parametric form, and this explains that in terms that someone who had only done regular algebra before could conceptualize.

[@Nhalok]

Thank you Firefly. You have helped me love maths again. A priceless gift. Thank you.

[@naderbelal8439]

I would like to thank you for this video ... it had helped me to solve a parametric geometrical problem

[@Observe3]

Your videos are flawless! you're a great asset to any 6th from student (me included :L) i only wish i found your videos sooner!! never stop making them man!! Big up yuhself

[@fireflylectures]

youragiantnoobfag - Thanks man. :)

[@kazooiebolt9473]

congrats on the name change from 7 years ago haha

[@treyfrost1059]

Best explanation I've found so far. Thanks

[@MLGJuggernautgaming]

Great advice and help! I found it much easier though to just take the cross product of the two planes' normal vectors and get a point on the line. It advice all the system of equations and possible mistake that can be made along the way.

[@aybarspektas1062]

That was crystal clear, thanks a lot!

[@Eddiecoolkid]

Very well explained - another little tool that I can use in solving vector questions. Thanks!

[@gurra1351]

Another easy way to do it would be to get the vectors of the two planes, n1 and n2. And by them create an orthogonal vector through cross product. So n1 x n2 = v or whatever you wanna call the cross vector. Since it's a line we "add" the vector with the corresponding point of the line. Side question, is it that important to get the point you got there or is it just adequate to get any arbitrary point on the line and then add the vector?

[@fireflylectures]

@Trey Frost - Thanks! Glad it helped. :)

[@santoshkumarmandal5822]

Firefly Lectures uu

[@forthrightgambitia1032]

For those who have done calc 3, in terms of the intersection I like to think of this as finding the levels curves (i.e. the 2-d) lines that are equal in each plane.

[@ronitmandal7301]

Thank you! This was the best method I found so far to solve these kinda problems

[@fonytas]

Clear explanation ! Thank you so much !

[@creampuff966]

Beautiful explanation! Thank you

[@surajkishan3837]

Thank you it really helped me

[@superminecraft01PvP]

thanks man, you're a rock star!

[@tuyatuguldur6362]

Thank you!!! I was struggling with vectors!!!

[@matthewpenner2348]

thanks! you really made it easy to understand!

[@Yarzongei]

Excellent, very helpful!

[@Dom-qo9cf]

thanks for the help! it was a good refresher

[@harsh.w0w643]

you can also put z = 0, solve for y and x (faster method)

[@luisrodriguez-mx7me]

can you also find the normal vectors, take the cross product, find a point of interception, then create a line out of the?

[@fireflylectures]

Yes, that will work as well!

[@philipgao7648]

+Firefly Lectures how so

[@jacobkuca7711]

exactly what he just said lol

[@marcoolariu5282]

Thanks so much for the reminder.

[@wilveryescas4166]

i read the whole chapter in the calc book and could not understand this problem but now thanks to you i get it thanks

[@josephmuriuki5147]

Thank you ;your video was very helpful in the topic.👏👏👏

[@HadiExtreme]

How would you find a line perpendicular to the line you found in this video given a point that line passes through?

[@heinrichkreuser2659]

Great video, thanks!

[@moamensmathsplatform-mmp6483]

Please would you please tell me what is the application which you used to record videos ?

[@marieclaireuwizeyimana7174]

Thank you very much for this easy method of finding line of intersection of two plane

[@andrewcarter215]

Thanks for the help!

[@nathankravitz8880]

Thanks man, very helpful

[@HamzaQayyum]

A much easier method to find the direction vector of the line of intersection would be to take the cross product of the normal vectors of both lines. From there, you can just generate a point using the plane equations and you have a line.

[@travisbickle5829]

Great video, thanks.

[@sumandas201]

Great sir love from India 👍👍

[@shahsingh1150]

Your lec is owsam .love❤ from india 🇮🇳

[@jacquelinelam1914]

this is so helpful thank u so much!!

[@zheb96]

awesome method, firefly lectures!

[@ivajgtz]

How did you get y=y?

[@ShubhraMahala]

What you mean anything cant be equal to itself

[@priyanshukhandelwal2908]

We can find the cross product of these two plane vectors(normal vectors) and then just put x or y or z = 0 and then we will have two equations (planes) in terms of x and y..(if we put z=0) from there we can find value of x, y.... Hence we will have a point (x, y, 0) on line of intersection and a vector parallel to line of intersection... Now we can easily find line of intersection

[@milanvegarmv]

Very useful. Thank you

[@anastasiaanautodidact9856]

Thank you! You rock!

[@gitaluitel665]

Teachers like you are rare

[@user-wi2um3it8d]

Thank you!

[@kiettran3900]

My question is that what is the condition of t in order the equation of the line has meaning. IS that right if all the value of t will be accepted???

[@dedricktaylor1248]

Thank you

[@MisterDoctorWaffles]

Thank you very much. Very epic

[@ph4s4]

this BOY just saved my life

[@leocadieploumen6713]

I have a question, why is y=y?, why is that obviously?

[@priyatoshbasu9301]

Here they did this to make a relation which would help them derive the direction cosine of the line by taking all the variables in terms of y.

[@DSsagar825]

Thank You ❤️

[@hloniphanimaluleke9159]

Thank You

[@beautifulcriminalsocynical7536]

Thank you so much !

[@BSMheybem]

Great from Addis Ababa University ,Ethiopia

[@tankkong4749]

using a matrix and putting into row echelon form is a better way tbh

[@Laggy_0]

Great video

[@carlitoslomboy74]

Is this the same answer for this question "how many regions will there be if 5 lines are drawn such that no three lines intersect at a point?" thank you.

[@vidushaniperera3073]

If we set the equations in terms of a different variable is the end answer the same ?or can it be a different set of parametric equations ?

[@Alex-bc3tt]

For a given curve, the parametric representation is not unique.

[@0001-exe]

THANK YOU!!

[@Gabtube252]

Thanks a lot

[@elizabethkim5623]

How would we do this using matrices?

[@Yahya._]

Thank you sir

[@aryanpandey1084]

Thankyou so much sir

[@unrmaestro]

thanks helped a lot appreciated

[@anderesten]

Hi, I´m having troubles with the following problem: Line given by planes a: x=1 ; and b: y+z=0 I need to find the direction vector of the line. If somebody can shed some light to this, I will appreciate.

[@yohannesbeneberu6781]

Find an equation for the plane that contains the line x=1+t,y=3t,z=2tand is parallel to the line of intersection of the planes -x+2y+z=0 and x+z=-1.

[@imashfernando9751]

Simply "thank you sur !!!!!!!"

[@VanillaBean15]

Why don't we use matrices for solving this system?

[@ayushi9199]

what is the intersection of the pink line and yellow plane?

[@stephenmuteti2705]

Congratulations......keep it up!!🆚

[@shreyywadhwa]

Thanks!

[@avikpoudel2433]

How y is function of y

[@picosdrivethru]

what college course typically covers this using vectors? I went through the linear algebra MIT course and it didn't... is it vector calculus? vector analysis? calculus 3 (multivariable calculus?) I need all kind of 3d geometric intersections, line plane line line (or closest point) plane plane etc

[@fireflylectures]

vector calculus, also known as Calculus 3.

[@picosdrivethru]

@@fireflylectures Awesome, yeah that's what I was thinking. Thank you so much!

[@naveendongre6180]

Thanks sir😊😊😊

[@solomon2747]

Thanks sir

[@mattheus5433]

thanks!

[@mohamedlashen6380]

Perfect

[@TheTheAndroid]

amazing :)

[@divyeshkumar5388]

Love from India ❤️

[@naveenr4153]

This is sooooo good.

[@neehar6409]

iit m?

[@muskanarora5560]

What would be the intersection of ax+by=0 and z=0

[@MagmaCubePvP]

does it work the same way if its a line?

[@fireflylectures]

Not sure I follow. You mean if you are trying to determine if/when two lines intersect instead of two planes? If so, no that would not work the same way. While planes are likely to intersect (the only way they wouldn't would be if they were parallel), two 3D lines will rarely intersect -- plus the equations are not written in the same form as equations of planes.

[@MagmaCubePvP]

Firefly Lectures thanks that what i was asking haha only if my teacher could teach me i wouldnt have to look through youtube 8th grade is SOOOOOOOOOOO hard xD

[@fireflylectures]

MagmaCubePvP Wow, calc III stuff in 8th grade -- nice job! You're well on your way. Keep up the good work!

[@MagmaCubePvP]

Firefly Lectures thanks man keep making videos your doing good with that greenscreen face cam :D

[@kyruke]

how did you run out of colors that fast, I only counted 4 you used.

[@pablobarba1543]

What if you are given the line and you have to find 2 planes?

[@marygold7055]

it will never happen..

[@MegamikazeMoriko]

i'll give a great hint for you guys point of intersection of two lines (1 dimension objects in 2 dimensional space. which intersects into a 0 dimensional object) in the solution, you have to substitute variables 2 times to solve for 2 variables line of intersection of two planes (2 dimensional objects in 3 dimensional space. which intersects into a 1 dimensional object) in the solution, you have to substitute variables 3 times to solve for 3 variables dont you notice a pattern? you could technically use this pattern for solving for Nth dimensional objects but those are impractical in our 3d universe

[@hangkertzy9722]

thankyouu

[@shahrahman7337]

If you take the cross product of the normal vectors to each plane, you get a vector parallel to the line of intersection of the two planes

[@fireflylectures]

Yes you do! Then you need a point on the line of intersection to get the equation of the line.

[@sidharthr6615]

I have seen people doing the " set z=0" thing... what's all that about??

[@Youngballer739]

What a genius

[@biankabaranova3922]

Hello. I love your video but I have a question. Why does y=y?

[@perlaramos8783]

This a probably really easy question, but i'm stuck. What about a line of intersection of the planes x+y+z = 1 and x-y+z=1??????

[@shubhammathur4887]

+Perla Ramos x=1-t, y=0, z=t

[@Alex-bc3tt]

@@shubhammathur4887 kindly explain how you were able to find this

[@Alphasone0212]

good sahit boss

[@kiddcamel]

I love you

[@airbornesnow4541]

why I paying my university thousands a semester when this guy explains it 100x better than my useless professor

[@ddlover911]

How do u know y is t

[@rakuuun4582]

I don't get it. Why is y = y? and Why are we allowed to replace y with t?

[@kylegrima3748]

how come y = y ?