Linear algebra Let ||u||_2, d_2(u,v), and theta_2(u,v) denote the Euclidean norm

Question

Linear algebra Let ||u||_2, d_2(u,v), and theta_2(u,v) denote the Euclidean norm, distance, and angle, respectively, associated with the dot product u middot v, and suppose ||u||, d(u,v), and theta(u,v) are the corresponding quantities associated with the inner product defined on R^3 by = 3 u_1 v_1 + 2 u_2 v_2 + 6 u_3 v_3. Find (a) (-1,0) middot (3,2), ||(-1,0)||_2, ||(3,2) ||_2, d_2((-1,0), (3,2)), and theta_2 ((-1, 0), (3, 2)). And for u = (-1,0,-3) and v = (3,2,1) in R^3, find (b) u middot u, u middot v, and v middot v, and use those values to find u middot (9 u + 2 v), ||u||_2, d_2(u,v), and theta_2 (u, v). (c)

Explanation / Answer

a) (-1 , 0).( 3, 2) = -1*3 +0*2 = -3

||1, 0||2 =sqrt(1^2 +0) = 1

|| (3,2)||2 = sqrt(3^2 +2^2) = sqrt13

d2( (-1, 0) , (3 , 2) ) = sqrt( ( 3+1)^2 + (2)^2) = sqrt(20)

theta2( (-1,0) , (3,2) )

cos(theta)2 = (-1,0).(3,2) /||-1,0||*||3,2||

= -3/1*sqrt13 = -0.83

theta2 = cos^-1(0.83) = 146.31deg

b) u.u = (-1, 0,-3).(-1, 0 , -3) = 1 +0 +9 = 10

u.v = (-1, 0 , -3)(3,2,1) = -3-3 = -6

v.v = (3,2,1)(3,2,1) = 9+4 +1 = 14

u.(9u+2v) = 9u.u +2u.v = 9*(10)+ 2*-6 = 90 -12 = 78

||u||2 = sqrt(1^2 +0 +3^2) = sqrt10

d2(u, v) = sqrt(4^2 +2^2 +4^2) = sqrt(16+4+16) =6

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