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 $\displaystyle \sqrt{0^2+ 1^2+ 1^2}= \sqrt{2}$ so a unit vector in that direction is $\displaystyle \left<0, \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right>$, the vector Jhevon uses. Multiply that vector by the length of 5i- 7j+ 6k which is $\displaystyle \sqrt{5^2+ 7^2+ 6^2}= \sqrt{110}$.
Of course, that vector multiplied by -1 gives a second, equally correct, answer.