u = ( -8 , -8)
v = (-1 , 2 )
the magnitude of vector projection of u onto v =
dot product of u and v over the magnitude of v = (u . v )/ ll v ll
ll v ll = √(-1² + 2²) = √5
u . v = ( -8 , -8) . ( -1 , 2) = -8*-1+2*-8 = -8
∴ (u . v )/ ll v ll = -8/√5
∴ the vector projection of u onto v = [(u . v )/ ll v ll] * [v/ ll v ll]
= [-8/√5] * (-1,2)/√5 = ( 8/5 , -16/5 )
The other orthogonal component = u - ( 8/5 , -16/5 )
= (-8 , -8 ) - ( 8/5 , -16/5 ) = (-48/5 , -24/5 )
So, u as a sum of two orthogonal vectors will be
u = ( 8/5 , -16/5 ) + (-48/5 , -24/5 )
