Line of intersection of two planes

[Krushnaraj Pandya]

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?

 

[Charles Link]

You can pick any $x$ you like, and then solve for $y$ and $z$ to get the point $a$.

 

[Krushnaraj Pandya]

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

 

[Krushnaraj Pandya]

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 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.

 

[Krushnaraj Pandya]

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]

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]

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

 

[Charles Link]

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]

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