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Question
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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