Intersection Line of 2 Planes - How to Find It - Step by Step Method & Explanation - Vector Equation

To find the line of intersection of two planes we calculate the vector product (cross product) of the 2 planes" normals. This gives us the direction vector of the line. We then find a point on the line by letting z = 0 in the Cartesian equation of the two planes' equations and solving simultaneously for x and y. Once we have both:
- a direction vector of the line
- a point on the line
we can write its vector equation.

[@luka4868]

I dont get it how are we 100% sure that there is a point in which the z of line is 0? What if the plane intersection(line) is normal to z axis and above it?(im thinking of z as height)

[@RadfordMathematics]

That’s a very fair question, many of my students ask me the same (and I once asked myself the same!)
If the line’s equation is such that it’s direction vector has a non-zero z component (if it did it would mean that the line runs parallel to the xy plane and has a constant z coordinate all along it’s length) then given the parameter (typically “lambda” or “Mu”) takes-on all real numbers: at one stage or another the line must go through the point with z coordinate equal to 0 (since the line is infinitely long).
Hope that helps a bit?

[@luka4868]

I get it now, thank you for the answer :)

[@nayunkim8510]

​@@RadfordMathematics Does it matter which x/y/z variable I choose? (Given that the direction vector does not have any zero components) How about when the direction vector is, say, (1, 0, 3)?

[@Urasaka]

@@RadfordMathematics I still don't get it honestly. couldn't there be a line perpendicular to the xz plane, with a constant z coordinate all along it's length, not intersecting with the x axis neither is it the y axis. how can it have a point on it of z coordinate equal to 0?

[@zainahmad1621]

​@@Urasaka line is guaranteed to pass through at least 2 of x, y or z at 0. If z component of direction is 0, just set x or y = 0 in step 3 then.

[@LevySkulk]

I don't often comment on math videos back thank you for this one. I was stuck so long on one of these questions, went through the chapter twice trying to figure out what I was missing.
My textbook has exclusively shown cross products between two vectors with their tails at the same point. It makes perfect sense that the cross product works for any two vectors but it just wasn't something I had considered. I try my best to think geometrically about these problems, but it's really hard too when the textbook spends so little time explaining a geometric point of view and assumes you'll just sus out the finer details on your own

[@RadfordMathematics]

Thanks for your comment @LevySkulk ! I’m really glad it helped! I try my best to illustrate how things work and “fit together” it’s truly rewarding when I get feedback like yours! So again: thank you! Wishing you all the best! John Radford.

[@vc9588]

Such a clear and straightforward video. Thank you!

[@MANOJKUMAR-yk8rc]

Thank you really very much. I was in pretty much confusion before viewing this video but now it’s crystal clear.

[@RadfordMathematics]

Thanks for your comment! It’s great to hear/read this video helped 😊

[@Lovealways11100]

Great video, explained the simplest way.✌🏻

[@marcialilianaperezgonzalez4881]

Thank you so much!! english is not even my native language and you explained it so clearly that I could understand :)

[@samtaylor2244]

Great video, made this amazingly simple!!

[@RadfordMathematics]

Thanks for your comment Sam and for your kind words!
I’m truly glad this video helped 😊
Take good care ✌️

[@ashleyashiku4604]

Thank you! This video was so concise, well-explained, and easy to grasp.

[@RadfordMathematics]

Thanks for your kind comment Ashley!
I’m really glad it helped 😊

[@substandardnerd]

Thanks. I needed help with my calculus homework.

[@sundaymanali5854]

this video is a life saver lol

[@babusseus1105]

Very coherent and nice explanation! I would mention that the planes can run pararelly to the xy plane, making the z = 0 method unusable.

[@yaboi7239]

if that is the case set either x or y equal to zero

[@Hamza-cx9zf]

Incredibly useful, thanks a bunch

[@RadfordMathematics]

Thanks for your comment Hamza, I really appreciate it 😊.
Truly glad it helped!
Take good care 🌱

[@joseblua6566]

Thanks a lot, really appreciate this.

[@RadfordMathematics]

Thanks for your comment Jose 😊
I’m truly glad this video helped!!
Take good care ✌️

[@nobanolife7215]

Thank you so much for the helpful video! the diagram really helped in my understanding.

[@RadfordMathematics]

Thanks for your comment 😊
I’m really glad this helped!!
Take good care ✌️

[@karlie7]

Thank you for the clear and simple explanations! Great tutorial

[@RadfordMathematics]

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[@vickysekhon4866]

short and sweet and very informative. To put it quite simply, "phenomenal".
-vicky

[@RadfordMathematics]

Thanks for such a kind comment Vicky!
Truly glad this video helped 😊

[@Techtalksdaily]

thanks a lot. Very consice and informative

[@RadfordMathematics]

Thanks for your comment and for your kind words 😊
I’m really glad this video helped!!
Take good care ✌️

[@sinxcosx-ly6fl]

such a piece of video , thank you

[@RadfordMathematics]

Thanks for your kind comment!
Really glad it helped 😊

[@RadfordMathematics]

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[@RadfordMathematics]

Thanks for your comment Preyasu, I really appreciate it!
Truly glad this video helped 😊.
Take good care ✌️

[@ashanmunasinghe00]

Thank you very much

[@ayomitundeekundayo2653]

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[@RadfordMathematics]

Thanks for your comment Ayomitunde!!!
I’m really glad this video helped 😊
Take good care ✌️

[@BirManyak.]

Thanks. Nice explanation 🙏

[@RadfordMathematics]

Thanks for your kind comment!
Really glad it helped 😊

[@sinless4088]

very easy to go through. cheers

[@RadfordMathematics]

Thanks for the comment!
I’m really glad it helped 😊

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Excellent !!!

[@RadfordMathematics]

Thanks for your comment @Hello World! I really appreciate it 😊
I’m truly glad this video helped!
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[@gcma1999]

Great video!

[@RadfordMathematics]

Thanks for your kind comment Gabriel! I’m really glad this video helped 😊
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[@satazer5824]

Perfect❤ Very clear

[@RadfordMathematics]

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I’m really glad this video helped 😊

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[@RadfordMathematics]

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[@RadfordMathematics]

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[@varunreddy2994]

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[@sonorousgaming7202]

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[@RadfordMathematics]

Thanks for your comment!
Really glad it helped 😊

[@Maul505]

Does it have to be z can I use y or x as zero?

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[@RadfordMathematics]

Glad it helped 😀

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[@RadfordMathematics]

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[@Borgilian]

But isn't z=0 pretty much a quick hack? I presume that you're assuming the planes to be created around the origin of the scene (0,0,0), but what if they're created way ahead on the positive or negative z axis? You'll never get z=0 in that scenario. Also, in the end (5, -2, 0) is assumed to be the origin of the line to which you add the direction vector scaled by your t component. What if the origin is displaced significantly? Even if the t component is correct, won't you get a line that's just parallel with the intersection of the planes and has the same size, but is not located exactly at the intersection?

[@RadfordMathematics]

Hi Borgilian, thanks for your comment 😊
The only scenario in which you wouldn’t have a z=0 is if the two planes are parallel to the cycle plane, in which case the variable a wouldn’t appear in the planes equations (since a would be constant).
As for the origin of the line, strictly speaking the line’s equation can be defined with any point along its length and a direction vector (a vector running parallel to the line) so (5,-2,0) definitely does the job 😊
Hope that helps.
Take good care ✌️

[@GreenMeansGOF]

@@RadfordMathematics That’s what I am confused about as well. In the case where z=0 does not work, does that mean that one of the variables is not present?

[@denver_dalman]

@@GreenMeansGOF I think there would always be z=0 as the intersecting line is not limited to the line showed in the illustration (if that confused you into thinking that lines and planes are limited to a certain range). Planes extends indefinitely, and so do lines, but in two directions, thus z=0 will always be part of the line.

[@gcma1999]

I was questioning that myself too. But when you remember that planes extend infinitely, it all becomes clear.

[@GeneTakavic]

why are you subtracting 1pi from 2pi? Doesn't make sense...

[@beanhead4197]

please explain like I am 3 years old

[@Deltax0428]

you can just simply put in equation these two planes and find the line passing the 2 planes !