Let P1 denote the equation of a plane to which the vector (i^+j^) is normal and which contains the line whose equation r→=i^+j^+k^+λ(i^−j^−k^) and P2 denote the equation of the plane containing the line L and a point with position vector j^ Which of the following holds good?
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a
The equation of x+y=2
b
The equation of r→⋅(i^−2j^+k^)=2
c
The acute angle between P1 and P2 is cot−1(3)
d
The angle between the plane P2 and the line L is tan−13
answer is A.
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Detailed Solution
Plane P1, contains the line r→=i^+j^+k^+λ(i^−j^−k^) hence contains the point i^+j^+k^ and is normal to vector (i^+j^)(r→−(i^+j^+k^))⋅(i^+j^)=0or x+y=2Plane P2, contains the liner→=i^+j^+k^+λ(i^−j^−k^) and point j^Hence, equation of plane is x−0y−1z−01−01−11−01−1−1=0or x+2y−z=2ff θ is the acute angle between P1 and P2,then cosθ=n1→⋅n2→|n→|n2→(i^+j^)⋅(i^+2j^−k^)2⋅6 =32⋅6=32θ=cos−132=π6 As L is contained in P2⇒θ=0