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# hw09 - v(1 v(2 and v(3 Solved Examples S 9.1 P 2.25 in...

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ENEE 241 02 * HOMEWORK ASSIGNMENT 9 Due Thu 03/12 Solve by hand without using calculator matrix functions. Show all intermediate steps. Let v (1) = [ 3 - 1 - 5 3 ] T , v (2) = [ - 1 - 1 6 - 4 ] T and v (3) = [ 13 - 7 1 1 ] T Consider the tetrahedron with vertices A 0 , A 1 , A 2 and A 3 , where A 0 is the origin (i.e., the all-zeros vector), and A 1 , A 2 and A 3 are respectively given by the points v (1) , v (2) and v (3) . (i) (3 pts.) Determine the projection f of v (2) onto v (1) . (ii) (3 pts.) What is the length of the error vector f - v (2) ? What is the area of the triangle A 0 A 1 A 2 ? (iii) (5 pts.) Determine the projection g of v (3) onto the plane defined by v (1) and v (2) . (iv) (4 pts.) What is the length of the error vector g - v (3) ? What is the volume of the tetrahedron A 0 A 1 A 2 A 3 ? ( Note: The volume of a tetrahedron is given by (Base Area)(Height)/3 .) (v) (5 pts.) Let s = [ 4 1 3 - 6 ] T . Compute the vector r = s + 1 4 v (1) - 1 2 v (2) - 1 4 v (3) What are the values of the inner products of r with each of the three v ( i ) ’s? Without solving a system of equations, determine the projection p of s on the subspace defined by the vectors
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Unformatted text preview: v (1) , v (2) and v (3) . Solved Examples S 9.1 ( P 2.25 in textbook). Consider the three-dimensional vectors v (1) = [-1 1 1 ] T , v (2) = [ 2-1 3 ] T , and s = [ 1 2 3 ] T . (i) Determine the projection ˆ s of s on the plane deﬁned by v (1) and v (2) . (ii) Show that the projection of ˆ s on v (1) is the same as the projection of s on v (1) . (Is this result expected from three-dimensional geometry?) S 9.2 ( P 2.26 in textbook). Let a (1) , a (2) , a (3) and a (4) be the columns of the matrix A = 1 1 / 2 1 / 4 1 / 8 1 1 / 2 1 / 4 1 1 / 2 1 Determine the least squares approximation p = c 1 a (1) + c 2 a (2) + c 3 a (3) of a (4) based on a (1) , a (2) and a (3) . Also determine the relative (root mean square) error k p-a (4) k k a (4) k...
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