If `vec(a) = hat(i) + hat(j) + hat(k)` and `vec(b) = hat(j) - hat(k)`, find a vector `vec(c )`, such that `vec(a) xx vec(c ) = vec(b)` and `vec(a).vec(c ) = 3`.

If vec(a) = hat(i) - hat(j) + 7hat(k) and vec(b) = 5hat(i) - hat(j) + lambda hat(k) , then find the value of lambda so that vec(a) + vec(b) and vec(a) - vec(b) are perpendicular vectors.

If vec(AB) = 3hat(i) + 2hat(j) + 6hat(k), vec(OA) = hat(i) - hat(j) - 3hat(k) , find the value of vec(OB) .

If vec(a) = 2hat(i) - 3hat(j) + 4hat(k), vec(b) = hat(i) + 2hat(j) - 3hat(k) and vec(c ) = 3hat(i) + 4hat(j) - hat(k) , then find vec(a).(vec(b) xx vec(c )) and (vec(a) xx vec(b)).vec(c ) . Is, vec(a).(vec(b) xx vec(c )) = (vec(a) xx vec(b)).vec(c ) ?

Let vec(a) = hat(i) + 4hat(j) + 2hat(k), vec(b) = 3hat(i) - 2hat(j) + 7hat(k) and vec(c )= 2hat(i) - hat(j) + 4hat(k) . Find a vector vec(p) which is perpendicular to both vec(a) and vec(b) and vec(p).vec(c ) = 18 .

Let vec(a) = hat(i) - 2hat(j) +3hat(k) . If vec(b) is a vector such that vec(a) . vec(b) = | vec(b)|^(2) and |vec(a) - vec(b)| and | vec(a) - vec(b)| = sqrt(7) , then |vec(b)| = ______

If vec(a) = 2hat(i) + 2hat(j) + 3hat(k), vec(b) = -hat(i) + 2hat(j) + hat(k) and vec(c )=3hat(i)+2hat(j) such that vec(a)+lambda vec(b) is perpendicular to vec(c) , then find the value of lambda .

If vec(a) = 3hat(i) - hat(j) and vec(b) = 2hat(i) + hat(j) - 3hat(k) then express vec(b) in the form of vec(b) = vec(b_(1)) + vec(b_(2)) where vec(b_(1))||vec(a) and vec(b_(2)) perpendicular to vec(a) .

If vec(a) = 2hat(i) - hat(j) + 3hat(k) and vec(b) = (6hat(i) + lambda hat(j) + 9 hat(k)) and vec(a) is parallel to vec(b) , find the value of lambda .

Let vec(a) =hat(i) +2hat(j) +hat(k),vec(b) =hat(i) -hat(j) +hat(k) and vec(c ) =hat(i) +hat(j) -hat(k) , vector in the plane of vec(a) and vec(b) whose projection on vec(c ) is 1/(sqrt(3)) is ______.