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Question

Find the equation of the line passing through the point (3,0,1) and parallel ti the planes x+2y=0 and 3y-z=0.

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Solution

Equation of the two planes are x+2y =0 and 3y-z=0.
Let n1 and n2 are the normals to the two planes, respectively.

n1=ˆi+2ˆj and n2=3ˆjˆk
Since, required line is parallel to the given two planes.
Therefore, b=n1×n2=∣ ∣ ∣ˆiˆjˆk120031∣ ∣ ∣
=ˆi(2)ˆj(1)ˆk(3)=2ˆi+ˆj+3ˆk
So, the equation of the lines through the point (3,0,1) and parallel to the given two planes are
(x3)i+(y0)j+(z1)k+λ(2ˆi+ˆj+3ˆk)
(x3)ˆi+yˆj+(z1)ˆk+λ(2ˆi+ˆj+3ˆk) = 0


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