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Question

Let P1 denote the equation of a plane to which the vector (^i+^j) is normal and which contains the line whose equation is 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?

A
The equation of P1 is x+y=2
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B
The equation of P2 is r.(^i2^j+^k)=2
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C
The acute angle between P1 and P2 is cot1(3)
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D
The angle between the plane P2 and the line L is tan13
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Solution

The correct option is C The acute angle between P1 and P2 is cot1(3)
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).

Hence, equation of plane is,
(r(^i+^j+^k)).(^i+^j)=0
x+y=2

Plane P2 contains the line r=^i+^j+^k+λ(^i^j^k) and point ^j Hence, equation of plane is ∣ ∣x0y1z0101110111∣ ∣=0
x+2yz=2

If θ is the acute angle between P1 and P2, then cos θ=n1.n2|n1||n2|=∣ ∣(^i+^j).(^i+2^j^k)26∣ ∣
=326=32
θ=cos132=π6
As L is contained in P2θ=0

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