Given \(\vec a=3\hat i+2\hat j+2\hat k\) and \(\vec b=\hat i+2\hat j-2\hat k\)
We need to find the vector perpendicular to both the vectors \(\vec a+\vec b\) and \(\vec a-\vec b\).
Recall a vector that is perpendicular to two vectors
Here, we have (a1, a2, a3) = (4, 4, 0) and (b1, b2, b3) = (2, 0, 4)
Let the unit vector in the direction of \((\vec a+\vec b)\times(\vec a-\vec b)\) be \(\hat p\).
We know unit vector in the direction of a vector \(\vec a\) is given by \(\hat a=\cfrac{\vec a}{|\vec a|}\).
Recall the magnitude of the vector \(\text x\hat i+y\hat j+z\hat k\) is
Thus, the required unit vector that is perpendicular to both \(\vec a\) and \(\vec b\) is \(\cfrac13(2\hat i-2\hat j-\hat k).\)