Vector Equation of Line of Intersection of Two Planes

How do we find a vector equation of line of intersection of two planes x-2y+z=0 and 3x-5y+z=4? We first want to find two points on the line of intersection, and the two points must lie on the plane. After that, we can use the two points to find the direction vector of the line, then use one of the points to write down the parametrization.

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[bluekeenhd]

if you have two equations that do not cancel out a number, can you multiply one equation by a constant (on both sides) to cancel out a variable?
Ex: eq1 --> -x-3y-4z = 2
eq2 --> 2x + 2y +2z = 5

[Ppooh]

i used a random y value to get point B and when i subtracted B-A to get the vector, it did not match to method 2. I used fractions and i am certain i havent miscalculated. What may be the issue then? great video by the way

[arjunarun]

Great video, just wanted to say if two lines aren't parallel it doens't mean they must intersect (0:52-1:00)

[erickpascual-bautista]

hey I have a question so I understand approach two how when you cross them you get a vector perpendicular to the two norm vectors of my plane which result in the direction vector of the line of intersection, however where do I get my point from? since you need a direction vector and a point on the line to create the vector equation. do I just use the norm vector as one of my points
so norm vector 1 was <3, -5, 1>
so my equation would be
<3, -5, 1> + <-3, -2, -2>t
then obviously multiply it out if you want the parametric form

[Alborzhakimi]

hello, thank you for your video. I had a question regarding your first approach. I already used your second approach and then attempted to check my answer for V by using the first approach. However, the Z did not cancel out so I had to multiply the equation of the second plane by -2 to get the Z to cancel. Next, I proceeded with your approach and got 2 points and subtracted them from each other. However, the answer did not end up the same as the answer of V with the first approach. Do you have any idea why this is? I could send you a picture of my work so you could check it out, if you would like. Thanks for your time.

[marthamaywhovier]

Does it matter which one of the planes I subtract ?

[jimmmy]

so am i free to pick whichever number ?

[wackaflacka]

Didn't much like the video, not enough explanation. Needed approach 2, but everything was super fast forwards, expecially with the n1 (multiplication sign) n2. I mean I understood it because I knew how to do it, but for the people that don't?

[richardsdiana]

Can I give a vector equation for the intersection between the two planar equations?