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 = 5i + j and w = -2i - j What does v_1 equal? -22/5 i - 11/5 j 3/5 i - 6/5 j 22/5 i + 11/5 j -3/5 i + 6/5 j What does v_2 equal? 22/5 i + 11/5 j -22/5 i - 11/5 j 3/5 i - 6/5 j -3/5 i + 6/5 j

Solution

1) v = < 5 , 1 >

w = < -2 , -1 >

v.w = -10 -1 = -11

|w|^2 = 5

v1 = v.w / |w|^2 <w>

v1 = -11 / 5 < -2 , -1 >

v1 = 22/5 i + 11/5 j

3rd option is correct

2) v2 = v - v1

= < 5 , 1 > - < 22/5 , 11/ 5>

= < 3/5 , -6/5 >

= 3/5 i - 6/5 j

3rd option is correct