InterviewSolution
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InterviewSolution
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(i) If C is a given non-zero scalar and `overset(to)(A) " and " overset(to)(B)` be given non-zero vectors such that `overset(to)(A) bot overset(to)(B)` then find the vectors `overset(to)(X)` which satisfies the equations `overset(to)(A) "."overset(to)(X) =c " and " overset(to)(A) xx overset(to)(X) = overset(to)(B)` (ii) `overset(to)(A)` vectors A has components `A_(1), A_(2) , A_(3) `in a right -handed rectangular cartesian coordinate system OXYZ. The coordinate system is rotated about the X-axis through an anlge `(pi)/(2)` . Find the components of A in the new coordinate system in terms of `A_(1),A_(2),A_(3)` |
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Answer» Correct Answer - `(i) vec(X) = ((vec(c))/(|vec(A)|^(2))) vec(A) - ((1)/(|vec(A)|^(2))) (vec(A) xx vec(B)) " " (ii) (A_(2) hat(i) -A_(1) hat(j) + A_(3) hat(k)) (i) Given `vec(A) bot vec(B) rArr vec(A) ". " vec(B)=0` 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 be represented as a linear combination of `vec(A) " and " vec(A) xx vec(B)` , Let us 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)` `rArr m = -(1)/(|vec(A)|^(2))` `:. vec(X) =((C)/(|vec(A)|^(2)))vec(A) -((1)/(|vec(A)|^(2))) (vec(A) xx vec(B))` (ii) Since vector `vec(A)` has components `A_(1) , A_(2) , A_(3)` in the coordinate system OXYZ `:. vec(A) = A_(1) hat(i) + A_(2) hat(j) +A_(3) hat(k)` When the given system is rotated about an angle of `pi//2` the new X-axis is along old Y-axis and new Y-axis is along the old negative X - axis , whereas z remains same . Hence the components of A in the new system are `(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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