Line of Intersection of Two Planes
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- Опубліковано 7 лют 2025
- If you enjoyed this video, take 30 seconds and visit fireflylecture... to find hundreds of free, helpful videos. No registration required! 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.
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
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!
@@Patrick_Knowlton I love this so much hahaha
Omg wow...glad to know this has practical use😊
@@Patrick_Knowlton oh wow , nice
@@Patrick_Knowlton did u get it right?
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.
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 :)
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
youragiantnoobfag - Thanks man. :)
congrats on the name change from 7 years ago haha
you can also put z = 0, solve for y and x (faster method)
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.
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
I would like to thank you for this video ... it had helped me to solve a parametric geometrical problem
Thank you Firefly. You have helped me love maths again. A priceless gift.
Thank you.
Thank you it really helped me
can you also find the normal vectors, take the cross product, find a point of interception, then create a line out of the?
Yes, that will work as well!
+Firefly Lectures how so
exactly what he just said lol
i read the whole chapter in the calc book and could not understand this problem but now thanks to you i get it thanks
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
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?
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.
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.
I have a question, why is y=y?, why is that obviously?
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.
That was crystal clear, thanks a lot!
Your lec is owsam .love❤ from india 🇮🇳
Does this work for all even with those eqn with coefficients? Cause I tried using same solving this getting different answers .... The line of intersection 5x-2y-2z=1 and 4x+y+z=6
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.
Great sir love from India 👍👍
Teachers like you are rare
How did you get y=y?
What you mean anything cant be equal to itself
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
vector calculus, also known as Calculus 3.
@@fireflylectures Awesome, yeah that's what I was thinking. Thank you so much!
thanks for the help! it was a good refresher
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.
why I paying my university thousands a semester when this guy explains it 100x better than my useless professor
Please would you please tell me what is the application which you used to record videos ?
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???
this BOY just saved my life
Thank you! This was the best method I found so far to solve these kinda problems
thanks! you really made it easy to understand!
Best explanation I've found so far. Thanks
Thank you ;your video was very helpful in the topic.👏👏👏
what is the intersection of the pink line and yellow plane?
Thank you very much for this easy method of finding line of intersection of two plane
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 ?
For a given curve, the parametric representation is not unique.
Thanks for the help!
Very well explained - another little tool that I can use in solving vector questions. Thanks!
Great video
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.
How would you find a line perpendicular to the line you found in this video given a point that line passes through?
Thank You ❤️
Excellent, very helpful!
Thank you very much. Very epic
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
Yes you do! Then you need a point on the line of intersection to get the equation of the line.
using a matrix and putting into row echelon form is a better way tbh
Great from Addis Ababa University ,Ethiopia
Love from India ❤️
@Trey Frost - Thanks! Glad it helped. :)
Firefly Lectures uu
thanks man, you're a rock star!
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.
Simply "thank you sur !!!!!!!"
Clear explanation ! Thank you so much !
Thankyou so much sir
Thank you sir
Why don't we use matrices for solving this system?
THANK YOU!!
Beautiful explanation! Thank you
Thank you!!! I was struggling with vectors!!!
I don't get it. Why is y = y? and Why are we allowed to replace y with t?
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
Thank you! You rock!
Great video, thanks!
Thanks man, very helpful
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??????
+Perla Ramos x=1-t, y=0, z=t
@@shubhammathur4887 kindly explain how you were able to find this
How would we do this using matrices?
does it work the same way if its a line?
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.
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
MagmaCubePvP Wow, calc III stuff in 8th grade -- nice job! You're well on your way. Keep up the good work!
Firefly Lectures thanks man keep making videos your doing good with that greenscreen face cam :D
awesome method, firefly lectures!
How y is function of y
Very useful. Thank you
Hello. I love your video but I have a question. Why does y=y?
This is sooooo good.
iit m?
this is so helpful thank u so much!!
Congratulations......keep it up!!🆚
What a genius
What if you are given the line and you have to find 2 planes?
it will never happen..
My question is to uh is....now we hv got X,Y,Z then wht...have we had to do further as defferentiation ???? Rply meh man...
Thanks so much for the reminder.
how did you run out of colors that fast, I only counted 4 you used.
Why does y = y?
I have seen people doing the " set z=0" thing... what's all that about??
Thanks a lot
Perfect
Thank you so much !
What would be the intersection of ax+by=0 and z=0
Thanks sir
Passed my exam, thanks!
That's what I dreamt when I feel asleep watching this video.
Thanks sir😊😊😊
Thank you
Thank you!
thanks helped a lot appreciated
Its basically intersection Q n P .
Let z=t basicall smalller and easier
Thanks!
How do u know y is t
I love you
good sahit boss
how come y = y ?