Question

If the vectors (sec² A) i + j + k,  i + (sec²B)j + k, i + j + (sec²C) k are coplanar, then find the value of  cosec²A  + cosec²B + cosec²C.

Answer 1

For three vectors to be coplanar we have

Which gives

 

Substituting equation 2 in 1 we have

So we have

= (x - 2) (y - 2) ( z - 2) - (x - 2) (y -1) (z -1) - (x - 1) (y - 2) (z - 1) - (x - 1) (y - 1) (z - 2) + 2(x - 1)(y - 1)(z - 1) = 0

Solving we have x + y + z = 4

Hence cosec²A + cosec²B + cosec²C = 4