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Question
Find the Cartesian equation of the plane passing through the points A(1, 1, 2), B(0, 2, 3) C(4, 5, 6)
Solution
If A(x1, y1, z1), B(x2, y2, z2) and C(x3, y3, z3) be three non-collinear points and P(x, y, z) be any point on a plane, then the Cartesian equation of the plane passing through A, B, C is
`|(x - x_1, y - y_1, z - z_1),(x_2 - x_1, y_2 - y_1, z_2 - z_1),(x_3 - x_1, y_3 - y_1, z_3 - z_1)|` = 0
∴ The Cartesian equation of the plane passing through A(1, 1, 2), B(0, 2, 3) and C(4, 5, 6) is
`|(x - 1, y - 1, z - 2),(0 - 1, 2 - 1, 3 - 2),(4 - 1, 5 - 1, 6 - 2)|` = 0
∴ `|(x - 1, y - 1, z - 2),(-1, 1, 1),(3, 4, 4)|` = 0
∴ (x – 1)(4 – 4) – (y – 1)(–4 – 3) + (z – 2)(–4 – 3) = 0
∴ 7y – 7 – 7z + 14 = 0
∴ y – z + 1 = 0
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