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Find a unit vector perpendicular to each of the vectors a + b and a - b, where a = 3i + 2j + 2k and b = i + 2j - 2k.

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 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  \(\cfrac13(2\hat i-2\hat j-\hat k).\)

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