Let u 1 37 v 0 42 and w 307 be row vectors with real entr

Let u - (1, -3,7), v - (0, -4,2), and w = (3,0,7) be row vectors with real entries. Find u v, ||w||, comp_uv, and the angle between v and w.

u = ( 1, -3 , 7) v = ( 0, -4 ,2) w= ( 3,0, 7)

Dot product u.v = 1*0 + (-3)*(-4) + 7* 2 = 12+14 = 26

Modulus || w|| = sqrt( 3^2 + 0^2 +7^2) = sqrt( 9 +49) = sqrt58

Comp_uv = |v| cos(theta) =( u.v)/|u|

|u| = sqrt( 1^2 + 3^2 +7^2) = sqrt(59)

|v| = sqrt( 0^2 + -4^2 + 2^2) = sqrt( 16+4) = sqrt20

Comp_uv = 26/sqrt59

Angle between v and w:

find dot product of v and w : v.w = 0*3 -4*0 + 2*7 = 14

cos(theta) = (u.w)/{ |u||w| } = 14/( sqrt(58)*sqrt(20) ) = 14/sqrt(58*20) =14/ 34.058 =0.411

theta = 65.72 deg angle between v and w