The value of a,b,c when `[(0,2b,c),(a,b,-c),(a,-b,c)]` is orthogonal , are :

Options

A. `1/2`
B. `1/3`
C. `1/6`
D. 1

Answer

C

Explanation

If A is an orthogonal matrix, then
`A A^T=I=A^TA`
`rArr{:[(0,2b,c),(a,b,-c),(a,-b,c)][(0,a,a),(2b,b,-b),(c,-c,c)]=[(1,0,0),(0,1,0),(0,0,1)]:}`
`rArr{:[(4b^2+c^2,2b^2-c^2,-2b^2+c^2),(2b^2-c^2,a^2+b^2+c^2,a^2-b^2-c^2),(-2b^2+c^2,a^2-b^2-c^2,a^2+b^2+c^2)]=[(1,0,0),(0,1,0),(0,0,1)]:}`
`rArr 4b^2+c^2=1,2b^2-c^2=0 and a^2-b^2-c^2=0`
`rArra^2=1/2,b^2=1/6and c^2=1/3rArrabs(abc)=1/6`