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प्रश्न
Find the Cartesian equation of the plane passing through the points A(1, 1, 2), B(0, 2, 3) C(4, 5, 6)
उत्तर
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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