Answer to Question #310257 in Analytic Geometry for tracerow01

Question

1.Consider the plane â„¿₁ : 3z + 2y + x = 2.

a.) If â„¿_{2 :} 2z - 3y = 1 is another plane. Are the planes â„¿₁ and â„¿₂ 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 â„¿₁ and the line L.

1]

Answer 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 )