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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)
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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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