Show that there are infinitely many vectors in R with Euclidean norm 1 whose Euclidean inner product with (1, –2,4) is zero.

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Solve number 18 please

Problem related to Euclidean Space and Linear Transformation:
Show that there are infinitely many vectors in R’with Euclidean norm 1 whose Euclidean inner
product with (1, –2,4) is zero.
18.
Find two vectors of norm 1 that are orthogonal to the vectors u =
w = (3,2,5,4).
(2,1, –4,0), v = (-1,-1,2,2), and
19.
Find the standard matrix for the linear operator T:R³ → R³ that first rotate a vector counterclockwise
about the y-axis through an angle 0 = 60, then reflect the resulting vector about the xz -plane, and
then transform the resulting vector by the formula T(x,y,z) = (2x – 3y + 4z, –3x + y – 2z,x +
3y). Hence compute T(1, –2,1).
20.
%3D
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Center of Excellence in Higher Education- North South University
Transcribed Image Text:Problem related to Euclidean Space and Linear Transformation: Show that there are infinitely many vectors in R’with Euclidean norm 1 whose Euclidean inner product with (1, –2,4) is zero. 18. Find two vectors of norm 1 that are orthogonal to the vectors u = w = (3,2,5,4). (2,1, –4,0), v = (-1,-1,2,2), and 19. Find the standard matrix for the linear operator T:R³ → R³ that first rotate a vector counterclockwise about the y-axis through an angle 0 = 60, then reflect the resulting vector about the xz -plane, and then transform the resulting vector by the formula T(x,y,z) = (2x – 3y + 4z, –3x + y – 2z,x + 3y). Hence compute T(1, –2,1). 20. %3D Page 3 of 3 Center of Excellence in Higher Education- North South University
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