Decompose v into two vectors v1 and v2 where v1 is parallel

Decompose v into two vectors v_1 and v_2, where v_1 is parallel to w and v_2 is orthogonal to w. v = -10i - 6j and w = i + j What does v_1 equal? -8i-8j -2i + 2j 2i - 2j 8i + 8j What does v_2 equal? -8i-8j 8i +8j -2i + 2j 2i - 2j

Solution

V2 = x+iy ; v1 = k(w) = k(i +j)

v2*w =0 ; x+y =0

(x+k) = -10 ; y +k = -6

x -y = -10 +6 = -4

x- y = -4

x +y =0

-------- x =-2 ; y = 2

k = -10 +2 = -8

v1 = -8i -8j ; v2 = -2i +2j