I have got this line g : x → = ( − 1 2 0...

Parker Bird

Parker Bird

Answered

2022-07-23

I have got this line
g : x = ( 1 2 0 ) + s ( 1 4 1 )
And this plane:
E t : t x + y + t z = 0
Now for which t is the line element of the plane?
My calculations: When I put the equation of the line in the equation of the plane i get: And this plane:
t ( 1 s ) + ( 2 + 4 s ) + t ( 2 ) = 0
This is the same as:
2 t + s ( 4 2 t ) = 0
=> s = t 2 4 2 t = 1 2
And now I do not know how to go on.
The solution is t=2
Can someone explain the last step to the solution please?

Answer & Explanation

kartonaun

kartonaun

Expert

2022-07-24Added 14 answers

The line and the plane are parallel iff the plane's normal vector (t,1,t) and the line's direction vector (−1,4,−1) are perpendicular; this gives t=2.
As for t=2 the vector (−1,2,0) lies in the plane the whole line lies in that plane.
Taniya Burns

Taniya Burns

Expert

2022-07-25Added 4 answers

Calling
p = ( x , y , z ) n = ( t , 1 , t ) p 0 = ( 1 , 2 , 0 ) v = ( 1 , 4 , 1 )
we have the plane Π p n = 0 and the line L p = p 0 + s v . Now if L Π we have
( p 0 + s v ) n = 0 { p 0 n = 0 v n = 0
From the second equation we have t + 4 t = 0 t = 2 and now making p 0 n = 2 × ( 1 ) + 1 × 2 + 2 × 0 = 0 hence for t = 2 L Π

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