InterviewSolution
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InterviewSolution
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(i) IfC isa givennon-zeroscalarand overset(to)(A)" and" overset(to)(B) be givennon-zerovectorssuch thatoverset(to)(A) bot overset(to)(B) then findthevectorsoverset(to)(X) whichsatisfies theequationsoverset(to)(A) "."overset(to)(X) =c" and" overset(to)(A) xxoverset(to)(X)= overset(to)(B) (ii) overset(to)(A) vectors A hascomponents A_(1), A_(2) , A_(3) in a right -handedrectangular cartesiancoordinate system OXYZ. Thecoordinate systemis rotated about theX-axis through an anlge(pi)/(2) . Findthecomponentsof Ain thenewcoordinatesystemin termsof A_(1),A_(2),A_(3) |
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Answer» and `vec(A) xx vec(X) =vec(B) rArr vec(A) ". " vec(B) =0 " and " vec(X)". " vec(B)=0` Now`[vec(X) vec(A) vec(A) xx vec(B)] = vec(X) ". " {vec(A) xx (vec(A) xx vec(B))}` ` =vec(X) .{(vec(A) ". " vec(B))vec(A)-(vec(A) ". " vec(B)) vec(B)}` ` = (vec(A) ". " vec(B))(vec(X) " ." vec(A)) - (vec(A) ". " vec(A)) (vec(X) " ." vec(B))=0` `rArr vec(X) , vec(A) , vec(A)xx vec(B)` are coplanar So `vec(X)` can berepresentedas alinearcombinationof `vec(A)" and"vec(A) xx vec(B)` , Letus consider, `vec(X) = lvec(A) + m (vec(A) xx vec(B))` Since `vec(A)" . " vec(X) = c` `:. vec(A) " ." {(vec(A) +m (vec(A)xx vec(B)) }=c` ` rArr l(vec(A) xx vec(A)) +m {vec(A) xx (vec(A) xx vec(B))}= vec(B)` `rArr 0- m |vec(A)|^(2)vec(B) =vec(B)` `rArrm = -(1)/(|vec(A)|^(2))` `:. vec(X) =((C)/(|vec(A)|^(2)))vec(A) -((1)/(|vec(A)|^(2))) (vec(A) xx vec(B))` (II) Sincevector`vec(A)`hascomponents `A_(1) , A_(2) , A_(3)` in thecoordinatesystemOXYZ `:. vec(A)= A_(1) HAT(i)+A_(2) hat(j)+A_(3) hat(k)` Whenthe givensystemis rotatedaboutan angleof `pi//2` the newX-axisis alongold Y-axisand newY-axisis alongthe oldnegativeX - axis, whereasz remainssame . Hencethe componentsof A in thenew systemare `(A_(2) , -A_(1), A_(3))` `:. vec(A) ` becomes`(A_(2) hat(i)- A_(2)hat(j)+ A_(3) hat(k))` |
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