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There are two vector `vec(A)=3hat(i)+hat(j)` and `vec(B)=hat(j)+2hat(k)`. For these two vectors- (i) Find the component of `vec(A)` along `vec(B)` and perpendicular to `vec(B)` in vector form. (ii) If `vec(A)` & `vec(B)` are the adjacent sides of parallelogram then find the magnitude of its area. (iii) Find a unit vector which is perpendicular to both `vec(A)` & `vec(B)`. |
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Answer» Correct Answer - (i) `1/5 (hat(j)+2hat(k)), 3hat(i)+4/5hat(j)-2/5hat(k)` (ii) 7 units (iii) `2/7hat(i)-6/7hat(j)+3/7 hat(k)` (i) component of `vec(A)` along `vec(B)=((vec(A).vec(B))/B)hat(B)` `=((vec(A).vec(B))/B)vec(B)/B=[((3hat(i)+hat(j)).(hat(j)+2hat(k)))/sqrt(5)]((hat(j)+2hat(k)))/sqrt(5)=1/5 (hat(j)+2hat(k))` Component of `vec(A) bot vec(B)` `=vec(A)-[(vec(A).vec(B))/vec(B)]hat(B)=3hat(i)+hat(j)-[1/5(hat(j)+2hat(k))]` (ii) Area of the parallelogram `=|vec(A)xxvec(B)|=|(hati,hatj,hatk) ,(3,1,0),(0,1,2)|=|2hati-6hatj+3hatk|` `=sqrt(2^(2)+(-6)^(2)+3^(2))=7` units (iii) Unit vector perpendicular to both `vec(A)` & `vec(B)` `hat(n)=(vec(A)xxvec(B))/(|vec(A)xxvec(B)|)=(2hati-6hatj+3hatk)/7=2/7hat(i)-6/7 hat(j)+3/7 hat(k)` |
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