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Question
Find the equation of the plane through the points (2, 1, 0), (3, –2, –2) and (3, 1, 7).
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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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