Line of intersection of planes is perpindicular to a line

[Faded-Maximus]

The problem is:

Show that the line of intersection of the planes 2x + y - 3z = 1 and -2x + 3y + 2z = 4 is perpendicular to the line r = (1, 3, 2) + t(2, -3, -2).

I have worked my way through this problem and I have the parametric equations for the line of intersection. I am stuck here and don't know how I can determine whether or not they are perpindicular.

Any help would be appreciated. Thanks.

Faded Maximus

 

[stapel]

Re: Line of intersection of planes is perpindicular to a lin

I have worked my way through this problem and I have the parametric equations for the line of intersection.

Please reply with this information. Thank you.

Eliz.

 

[Faded-Maximus]

I got the parametric equations by letting x = t and eliminating y and z in each case. We aren't allowed to use matrices for these questions, so I had to do it the other way.

By eliminating y I got z = -1-8t / -11
By elminiating z I got y = 14 + 2t / 11
and of course x = t.

If you want to see all my math to get to that point I will scan it.

 

[Denis]

I got the parametric equations by letting x = t and eliminating y and z in each case. We aren't allowed to use matrices for these questions, so I had to do it the other way.
By eliminating y I got z = -1-8t / -11
By elminiating z I got y = 14 + 2t / 11
and of course x = t.

2x + y - 3z = 1
-2x + 3y + 2z = 4 ; from these 2 equations:
4y - z = 5
z = 4y - 5

Using your equation z = -1 -8t / -11, then:
-1 -8t / -11 = 4y - 5
11 - 8t = -44y + 55
11y = 2t + 11
y = 2t / 11 + 1

you have: y = 2t / 11 + 14

One of us "goofed".

 

[pka]

The direction vector of the line of intersection is the cross product of the normals:

\(\displaystyle < 2,1, - 3 > \times < 2, - 3, - 2 > = < - 11, - 2, - 8 >.\)

The dot product of the direction vector and the intersection vector is:

\(\displaystyle < - 11, - 2, - 8 > \cdot < 2, - 3, - 2 > = ( - 11)(2) + ( - 2)( - 3) + ( - 8)( - 2) = 0.\)

Therefore the two lines are perpendicular, if they intersect.
So you must show that they do intersect!