Question

(1 point) Find two vectors vi and v2 whose sum is (2, -5), where vi is parallel to (3,-1) while v2 is perpendicular to (3,-1). V1 and

Answer 1

Let the two vector be v_1 = (a, b) v_2 = (c, d) v_1 + v_2 = (2, -5) implies (a, b) + (c, d) = (2, -5) i. e. a + c = 2 implies c = 2 - a ..... (i) b + d = -5 implies d = -5 -b....(ii) Since v_1 is parallel to (3, -1) Therefore v_1 = k (3, 1) for some k elementof R implies (a, b) = (3k, -k) implies a = 3k b = -k} ---(iii) using (iii) in (i) & (ii) c = 2 - 3k d = k -5}---(iv) Also v_2 is perpendicular to (3, -1) Therefore v_2 (3, -1) = 0 implies (c, d) middot (3, -1) = 0 implies 3c - d = 0 .... (v) putting the value of c, d from (iv) to (v) 3 (2 - 3k) - (k - 5) = 0 implies 6 - 9k - k + 5 = 0..