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Question
Find the equation of the plane through the points (2, 1, 0), (3, –2, –2) and (3, 1, 7).
Solution
Since, the equation of the plane passing through the points (x1, y1, z1), (x2, y2, z2) and (x3, y3, z3) 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
⇒ `|(x - 2, y - 1, z - 0),(3 - 2, -2 - 1, - 2 - 0),(3 - 2, 1 - 1, 7 - 0)|` = 0
⇒ `|(x - 2, y - 1, z),(1, -3, -2),(1, 0, 7)|` = 0
⇒ `(x - 2)|(-3, -2),(0, 7)| -(y - 1)|(1, -2),(1, 7)| + z|(1, -3),(1, 0)|` = 0
⇒ (x – 2)(– 21) – (y –1)(7 + 2) + z(3) = 0
⇒ – 21(x –2) – 9(y – 1) + 3z = 0
⇒ – 21x + 42 – 9y + 9 + 3z = 0
⇒ – 21x – 9y + 3z + 51 = 0
⇒ 7x + 3y – z – 17 = 0
Hence, the required equation is 7x + 3y – z – 17 = 0.
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