Find b and c so that (7, b, c. is orthogonal to both (1, 2, 3) and (1, −2, 1).

The cross product is perfect for this purpose.
The cross product C of vectors p and q is orthogonal to both p and q.
C will then be scaled to have the first value equal to 7 to find b and c.

C=<1,2,3>x<1,-2,1>
=
i j k
1 2 3
1 -2 1
=<2*1+3*2, -(1*1-1*3), 1*(-2)-1*2>
=<8, 2, -4>
=2<4,1,-2>
=(8/7)<4*7/4, 1*7/4, -2*7/4>
=(8/7)<7, 7/4, -7/2>

So b=7/4, c=-7/5

Check:
<7,7/4,-7/2>. <1,2,3>=7*1+7/4*2-7/5*3=7+7/2-21/2=0 ok, orthogonal
<7,7/4,-7/2>.<1,-2,1>=7*1-7/4*2-7/2*1=0 ok, orthogonal