Answer: The answer is 3.75 and -2.5.
Step-by-step explanation: We are given to find the values of b and c if the vector (5, b, c) is orthogonal to (1, 2, 3) and (1, -2, 1).
We know that if two vectors are orthogonal, then their dot product is equal to zero. So, we have
\((5, b, c).(1,2,3)=0\\\\\Rightarrow 5+2b+3c=0\\\\\Rightarrow 2b+3c=-5,~~~~~~~~~~~~~~~~~~~~(A)\)
and
\((5,b,c).(1,-2,1)=0\\\\\Rightarrow 5-2b+c=0\\\\\Rightarrow -2b+c=-5.~~~~~~~~~~~~~~~~~~~~~(B)\)
Adding equations (A) and (B), we have
\(3c+c=-5-5\\\\\Rightarrow 4c=-10\\\\\Rightarrow c=-2.5,\)
and from (B), we have
\(-2b+2.5=-5\\\\\Rightarrow -2b=-7.5\\\\\Rightarrow b=3.75.\)
Thus, the value of b is 3.75 and the value of c is -2.5.
