Write the vector u as a sum of two orthogonal vectors, one of which is the vector projection of u onto v

u = (-8, - 8), v = (-1, 2)

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)