Let II be the plane consisting of all points (x, Y, z) such that (x, y, z) = (3, 5, 0) + s(2,2, 4) + t(1, 3, –2) where s, t e R. (a) Find the Cartesian equation of the plane E such that E is parallel to II andE is a subspace of R³. (b) Find the orthogonal projection of (1, -2,2) onto the normal vector n to II. (c) Find the vector equation for the image of II under the linear transformation T: R3 → R³ given by T(x, y, z) = (x + y + z, y – 2z, 3x + 2).
Let II be the plane consisting of all points (x, Y, z) such that (x, y, z) = (3, 5, 0) + s(2,2, 4) + t(1, 3, –2) where s, t e R. (a) Find the Cartesian equation of the plane E such that E is parallel to II andE is a subspace of R³. (b) Find the orthogonal projection of (1, -2,2) onto the normal vector n to II. (c) Find the vector equation for the image of II under the linear transformation T: R3 → R³ given by T(x, y, z) = (x + y + z, y – 2z, 3x + 2).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Let II be the plane consisting of all points (x, Y, z) such that
(x, y, z) = (3, 5, 0) + s(2,2, 4) + t(1, 3, –2) where s, t e R.
(a) Find the Cartesian equation of the plane E such that E is parallel to II andE is a subspace
of R³.
(b) Find the orthogonal projection of (1, –2, 2) onto the normal vector n to II.
(c) Find the vector equation for the image of II under the linear transformation T : R3 → R³
given by
T(x, y, z) = (x + y +z, y – 2z, 3x+ 2).
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