1.Consider the plane ℿ1 : 3z + 2y + x = 2.
a.) If ℿ2 : 2z - 3y = 1 is another plane. Are the planes ℿ1 and ℿ2 orthogonal?
b.) Consider the line L that passes through the point P0(0, 1, 1) and is parallel to the vector
⟶
u = [ 1
1 Find the point of intersection of the plane ℿ1 and the line L.
1]
a) direction ratio of plane Π1 :
(1,2,3)
Direction ratio of plane Π2 :
(0,-3,2)
For orthogonal , dot product of direction ratio is equal to 0 .
1(0)+2(-3)+3(2) = 0-6+6 =0
Hence , planes are orthogonal .
b) line L has direction ratio equal to vector u I.e. =(1,1,1) and it passes to point (0,1,1)
So, equation of line is -
(x-0)/1 = (y-1)/1 = (z-1) /1
x = y-1 = z- 1 = w(let) ...........(i)
For point of intersection of plane and line,
x+ 2y+3z =2
w +2(w+1) + 3(w+1) =2
w + 2w +2 +3w +3 =2
6w +5 =2
6w = -3
w = -1/2
From equation (i) ,
x = w = -1/2
y = w+1 = -1/2 +1 = 1/2
z= w+1 = -1/2 +1 = 1/2
So, the point of intersection is
( -1/2 , 1/2 , 1/2 )
Comments
Leave a comment