Consider the plane: E : r = − 1 1 1 + λ 1 2 0 + μ 1 0 1 Let F be the plane containing the point A ( − 4 , 2 , 2 ) and parallel to E . Suppose point B is on plane E such that B has a

Question

Consider the plane: E : r = ​ − 1 1 1 ​ ​ + λ ​ 1 2 0 ​ ​ + μ ​ 1 0 1 ​ ​ Let F be the plane containing the point A ( − 4 , 2 , 2 ) and parallel to E . Suppose point B is on plane E such that B has a minimum distance from point A . If C ( − 3 , 0 , 4 ) lies in plane F , find the area of triangle A BC .

TestHub · Accepted Answer

Project A onto E to get B. AC lies in the parallel plane F and is perpendicular to AB. $n = (1, 2, 0) \times (1, 0, 1) = (2, - 1, - 2), ∣ n ∣ = 3. E : 2 x - y - 2 z = - 5. A = (- 4, 2, 2), n \cdot A = - 14. B = A - \frac{- 14 + 5}{9} n = A + n = (- 2, 1, 0) . A B = 3, A C = (1, - 2, 2), ∣ A C ∣ = 3, A C \cdot n = 0. Area = \frac{1}{2} (3) (3) = \frac{9}{2} = 4.5.$

Mathematics - 3D-Coordinate Geometry Question with Solution | TestHub

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