find the value of b if the resultant of 2i+(3b+1)j+3k and 2i+(2b+2)j+3k is orthogonal to the resultant of -2i+(b+1)j-k and -i +(2b-i)j+2k

Question

Orthogonal /perpendicular

Answer 1

The resultant of the first two vectors is 4i+(5b+3)j+6k; and the second two: -3i+3bj+k, assuming (2b-i) should be (2b-1).

The two resultants are orthogonal which means their dot product (scalar product)=0:

-12+3b(5b+3)+6=0; 15b^2+9b-6=0; 5b^2+3b-2=0=(5b-2)(b+1), so b=-1 or 2/5.