Let u 821 Find the decomposition u ab where vector a is pa

Let u = (8,2,1). Find the decomposition u = a+b, where vector a is parallel to v = (2,1,-1) and b is perpendicular to v.

Solution

Given that U = < 8, 2 ,1 > and V = < 2 , 1, -1 >

Now we find the decomposition U = a + b where vector a is parallel to V and b is perpendicular to V.

That is a = proj_v U = [ ( U . V ) / |V|² ] V = [ (< 8, 2 ,1 > . < 2 , 1, -1 > ) / | < 2 , 1, -1 >|² ] < 2 , 1, -1 >

                                                                   = (17/6) < 2 , 1, -1 > = < 17/3 ,   17/6,   -17/6 >

             b = U - proj_v U = < 8, 2 ,1 > - < 17/3 ,   17/6,   -17/6 >   = < 7/3 , - 5/6 ,   23/6 >

Thus the required decomposition is U = a + b = < 17/3 ,   17/6,   -17/6 > + < 7/3 , - 5/6 ,   23/6 >