determine a vector in the yz plane that is equally inclined to both the y and z axes and has the same length as p= 5i-7j+6k
I don't see how that is in any sense a "trick" question, unless you mean by that a question so simple you think there must be some hidden trick.
One "vector in the yz plane that is equally inclined to both the y and z axes" is, of course, <0, 1, 1>. That has length so a unit vector in that direction is , the vector Jhevon uses. Multiply that vector by the length of 5i- 7j+ 6k which is .
Of course, that vector multiplied by -1 gives a second, equally correct, answer.