219
Question
The vector equation of the plane passes through the points A&B with position vector 2^i+^j−^k & −^i+3^j+4^k respectively & ⊥er to the plane ¯¯¯r.(^i−2^j+4^k)=10 is
The vector equation of the plane passes through the points A&B with position vector 2^i+^j−^k & −^i+3^j+4^k respectively & ⊥er to the plane ¯¯¯r.(^i−2^j+4^k)=10 is
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Solution
The correct option is A ¯¯¯r.(18^i+17^j−3^k)=49
The required plane passes through the points
A(2^i+^j−^k) and B(−^i+3^j+4^k)
∴¯¯¯¯¯¯¯¯AB=¯¯¯¯¯¯¯¯OB−¯¯¯¯¯¯¯¯OA=−3^i+2^j+5^k
Again the required, plane is ⊥er to the plane
¯¯¯r⋅(^i−2^j+4^k)=10
∴¯¯¯n1=^i−2^j+4^k Let ¯¯¯n be the normal to the desired plane, then
¯¯¯n=¯¯¯n1ׯ¯¯¯¯¯¯¯AB=∣∣
∣∣ijk1−24−325∣∣
∣∣=−18^i−17^j−4^k
Now equation of the plane through A(2^i+^j−^k) &
normal to vector ¯¯¯n is ¯¯¯r⋅¯¯¯n=¯¯¯a⋅¯¯¯n
⇒¯¯¯r⋅(−18^i−17^j−4^k)=(2^i+^j−^k)⋅(−18^i−17^j−4^k)
⇒¯¯¯r⋅(18^i+17^j+4^k)=49
Hence Choice (A) is correct.
The required plane passes through the points
A(2^i+^j−^k) and B(−^i+3^j+4^k)
∴¯¯¯¯¯¯¯¯AB=¯¯¯¯¯¯¯¯OB−¯¯¯¯¯¯¯¯OA=−3^i+2^j+5^k
Again the required, plane is ⊥er to the plane
¯¯¯r⋅(^i−2^j+4^k)=10
∴¯¯¯n1=^i−2^j+4^k Let ¯¯¯n be the normal to the desired plane, then
¯¯¯n=¯¯¯n1ׯ¯¯¯¯¯¯¯AB=∣∣ ∣∣ijk1−24−325∣∣ ∣∣=−18^i−17^j−4^k
Now equation of the plane through A(2^i+^j−^k) &
normal to vector ¯¯¯n is ¯¯¯r⋅¯¯¯n=¯¯¯a⋅¯¯¯n
⇒¯¯¯r⋅(−18^i−17^j−4^k)=(2^i+^j−^k)⋅(−18^i−17^j−4^k)
⇒¯¯¯r⋅(18^i+17^j+4^k)=49
Hence Choice (A) is correct.
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