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 ∣∣
∣∣x−2y−2z+13−24−22+17−20−26+1∣∣
∣∣=0⇒∣∣
∣∣x−2y−2z+11235−27∣∣
∣∣=0⇒(x−2)(14+6)−(y−2)(7−15)+(z+1)(−2−10)=0⇒5x+2y−3z=17
The vector equation of the given plane is →r.(5^i+2^j−3^k)=17 The equation of the plane which is parallel to the above plane is given by →r.(5^i+2^j−3^k)=λ Since, it passes through(4,3,1)(4^i+3^j+^k)⋅(5^i+2^j−3^k)=λ⇒λ=23 Therefore, the equation of the required plane is →r.(5^i+2^j−3^k)=23