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Question

Find the vector and Cartesian equations of the plane passing through the points (2, 2, 1), (3, 4, 2) and (7, 0, 6). Also find the vector equation of a plane passing through (4, 3, 1) and parallel to the plane obtained above.

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Solution

The equation of the plane which passes through given points is given by
∣ ∣x2y2z+132422+172026+1∣ ∣ = 0∣ ∣x2y2z+1123527∣ ∣ = 0(x2)(14+6)(y2)(715)+(z+1)(210) = 05x+2y3z = 17

The vector equation of the given plane is r.(5^i+2^j3^k)=17
The equation of the plane which is parallel to the above plane is given by
r.(5^i+2^j3^k) = λ
Since, it passes through (4,3,1)(4^i+3^j+^k)(5^i+2^j3^k)=λλ = 23
Therefore, the equation of the required plane is r.(5^i+2^j3^k) = 23

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