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Find a unit vector perpendicular to both...

Find a unit vector perpendicular to both the vectors `(2hat(i)+3hat(j)+hat(k))` and `(hat(i)-hat(j)-2hat(k))`.

Text Solution

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Let `vec(A)=2hat(i)+3hat(j)+hat(k)` and `vec(B)=hat(i)-hat(j)+2hat(k)`
Unit vector perpendicular to both `vec(A)` and `vec(B)` is `hat(n)=(vec(A)xxvec(B))/(|vec(A)xxvec(B)|)`
`vec(A)xxvec(B)=|(hat(i),hat(j),hatk),(2,3,1),(1,-1,2)|=hat(i)(6+1)-hat(j)(4-1)+hat(k)(-2-3)=7hat(i)-3hat(j)-5hat(k)`
`:.|vec(A)xxvec(B)|=sqrt(7^(2)+(-3)^(2)+(-5)^(2))=sqrt(83)` unit
`:. hat(n)=1/sqrt(83)(7hat(i)-3hat(j)-5k)`
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