# Line of intersection of two planes

Gold Member

## Homework Statement

Find equation of line of intersection of planes r.(3i-j+k)=1 and r.((i+4j-2k)=2

## Homework Equations

a x b gives a perpendicular vector to a and b...(i)

## The Attempt at a Solution

to write equation of a line r=a+tb we need a point a on the line and a parallel vector b. Taking the cross product of 3i-j+k and i+4j-2k gives us b, assuming r to be xi+yj+zk, we get 3x-y+z=1 and x+4y-2z=2, putting...2 equations in three variables, how do we end up finding a and is there a faster method?

• Homework Helper
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2020 Award
You can pick any ## x ## you like, and then solve for ## y ## and ## z ## to get the point ## a ##.

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You can pick any ## x ## you like, and then solve for ## y ## and ## z ## to get the point ## a ##.
So my method is correct? That's basically what I did but I picked z=0 and found x and y

• Gold Member
You can pick any ## x ## you like, and then solve for ## y ## and ## z ## to get the point ## a ##.
so I got 14/39 , 5/13, 0 as a. r=a+t(-2i+7j+13k) is this a correct final answer

ehild
Homework Helper

## Homework Statement

Find equation of line of intersection of planes r.(3i-j+k)=1 and r.((i+4j-2k)=2

## Homework Equations

a x b gives a perpendicular vector to a and b...(i)

## The Attempt at a Solution

to write equation of a line r=a+tb we need a point a on the line and a parallel vector b. Taking the cross product of 3i-j+k and i+4j-2k gives us b, assuming r to be xi+yj+zk, we get 3x-y+z=1 and x+4y-2z=2, putting...2 equations in three variables, how do we end up finding a and is there a faster method?
As you wrote, you have two equations 3x-y+z=1 and x+4y-2z=2 and want the solution.You can choose one variable arbitrarily, as parameter. Say, x=t. Write y and z in terms of t and you have the parametric equation of the line of intersection.

• Gold Member
As you wrote, you have two equations 3x-y+z=1 and x+4y-2z=2 and want the solution.You can choose one variable arbitrarily, as parameter. Say, x=t. Write y ynd z in terms of t and you have the parametric equation of the line of intersection.
Seems like I'm on the right track. Thank you :D

SammyS
Staff Emeritus
Homework Helper
Gold Member

## Homework Statement

Find equation of line of intersection of planes r.(3i-j+k)=1 and r.((i+4j-2k)=2

## Homework Equations

a x b gives a perpendicular vector to a and b...(i)

## The Attempt at a Solution

to write equation of a line r=a+tb we need a point a on the line and a parallel vector b. Taking the cross product of 3i-j+k and i+4j-2k gives us b, assuming r to be xi+yj+zk, we get 3x-y+z=1 and x+4y-2z=2, putting...2 equations in three variables, how do we end up finding a and is there a faster method?
For equations:
##3x-y+z=1##
##x+4y-2z=2##​
Set ##\ x = 0\,, ## then divide the second equation by 2 & add the equations.

• Krushnaraj Pandya
Gold Member
For equations:
##3x-y+z=1##
##x+4y-2z=2##​
Set ##\ x = 0\,, ## then divide the second equation by 2 & add the equations.
Got it! Thank you :D

Homework Helper
Gold Member
2020 Award
Just to add on to what I think has already been said, the solution for a line is often written in the form ## \frac{x-x_1}{A}=\frac{y-y_1}{B}=\frac{z-z_1}{C} ## which can be set equal to ## t ##. If you pick ## x_1=0 ## it simplifies the first term. You can even multiply out the ## A ## and then set ## x=t ## as @ehild suggested.

• Krushnaraj Pandya
Gold Member
Just to add on to what I think has already been said, the solution for a line is often written in the form ## \frac{x-x_1}{A}=\frac{y-y_1}{B}=\frac{z-z_1}{C} ## which can be set equal to ## t ##. If you pick ## x_1=0 ## it simplifies the first term. You can even multiply out the ## A ## and then set ## x=t ## as @ehild suggested.
Crystal clear now! Thank you very much :D

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