# Find a Unit Vector Perpendicular to Both the Vectors → a and → B , Where → a = ˆ I − 7 ˆ J + 7 ˆ K and → B = 3 ˆ I − 2 ˆ J + 2 ˆ K . - Mathematics

Sum

Find a unit vector perpendicular to both the vectors veca and vecb , where veca = hat i - 7 hatj +7hatk  and  vecb = 3hati - 2hatj + 2hatk .

#### Solution 1

veca = hat i - 7 hatj +7hatk    vecb = 3hati - 2hatj + 2hatk .

perpandicular vector to both veca  &  vecb   "is"    vecc

hati = |[hati,hatj,hatk],[1,-7,7],[3,-2,2]|

= hati(0) - hatj(2-21)+hatk(-2+21)

= 0hati + 19hatj + 19hatk

⇒ vecc = 0hati + 19hatj + 19hatk

hatc = vecc/|vecc| = (0hati + 19hatj + 19hatk)/sqrt(0^2+19^2+19^2) = (19(hatj+hatk))/(19sqrt(2))

= (hatj + hatk)/sqrt(2)

vecc = 1/sqrt(2)(hatj+hatk)

#### Solution 2

veca = hat"i" - 7hat"j" + 7hat"k" and vecb = 3hat"i" - 2hat"j" + 2hat"k"

let vecn be the vector perpendicular to veca  "and"  vecb

vecn = veca xx vecb

vecn = |(hat"i", hat"j" ,hat"k") ,(1,-7,7),(3,-2,2)| = 19hat"j" + 19hat"k"

Now, the unit vector perpendicular to veca  "and"  vecb

hatn = (19hat"j" + 19hat"k")/sqrt(19^2 + 19^2) = 1/sqrt(2)(hat"j" + hat"k")

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