Explanation:Let us first find the angle between the two given VECTORS - Cos(A)=(i+j)∙(3i+4j)√[12+12]√[32+42]   Cos(A)=75√2   Sin2(A)=1−Cos2(A)=1−4950   Sin(A)=15√2  …. (Considering the acute angle, so taken only +ve value) Now let  r=3i+4j . This vector has a magnitude of  |r|=  √[32+42]=5 . If we RESOLVE  r  into two components, one parallel to  i+j  and other PERPENDICULAR to  i+j , then since the acute angle between these two given vectors is  A , so the magnitude of parallel component is  |r|  Cos(A)  and the perpendicular component is  |r|  Sin(A)   Since the QUESTIONS demands the perpendicular component, we need to multiply this magnitude with a UNIT vector perpendicular to  i+j , which is −  1√2i+1√2jSo the required answer is - |r|Sin(A)(−1√2i+1√2j)   =515√2(−1√2i+1√2j)   =−12i+12j   The question demand that the result must be in the same plane as  3i+4j , which it is given that the entire problem and solution is present only in 2D - X-Y plane. Let us check. If the answer is indeed perpendicular to  i+j , their dot product must be zero (−12i+12j)∙(i+j)   −12+12=0