Given vectors a vector and b vector the vector a vector can

Given vectors a vector and b vector, the vector a vector can be decomposed into the sum a vector = a vector_||_b vector + a vector_ _b vector where a vector_||_b vector = proj_b vector (a vector) and a vector_ b vector =orth_b vector (a vector). Find the decomposition a vector = a vector_||_b vector + a vector_ _b vector for a vector = (- 3, 2) and b = (1, 4). Then sketch all four vectors.

Solution

We have proj_b(a) = [(a.b)/(b.b)]b =[(-3*1+2*4)/(1+16)](-3,2) = (5/17)(-3,2) = (-15/17,10/17). Then orth_b(a) = a- proj_b(a) = (-3,2) –(-15/17,10/17)= (-36/17, 24/17). Thus, the required decomposition is (-3,2)= (-15/17,10/17)+ (-36/17, 24/17).