Consider two vectors A with rightwards harpoon with barb upwards on top equals A subscript x times i with hat on top plus A subscript y times j with hat on top plus A subscript z times k with hat on top and B equals B subscript x times i with hat on top plus B subscript y times j with hat on top plus B subscript z times k with hat on top. Match each of the following expressions for the coefficients of the vector product A with rightwards arrow on top cross times B with rightwards arrow on top with its correct unit vector: - plus open parentheses A subscript y B subscript z minus A subscript z B subscript y close parentheses - minus open parentheses A subscript x B subscript z minus A subscript z B subscript x close parentheses - plus open parentheses A subscript x B subscript y minus A subscript y B subscript x close parentheses a. i with hat on top
b. k with hat on top
c. j with hat on top

we have been given a quadratic function and we need to restrict the domain such that it becomes a one to one function.

we know that vertex of this quadratic function occurs at (5,2).

further, we know that range of this function is .

if we restrict the domain of this function to either or , it will become one to one function.

let us know find its inverse.

upon interchanging x and y, we get:

let us now solve this function for y.

hence, the inverse function would be if we restrict the domain of original function to and the inverse function would be if we restrict the domain to .

answer: the potential energy decreases from 379 j to 243 j

explanation:

for the law of conservation of energy, the total energy in a system must be conserved. the total energy of the system is given by the sum of kinetic (k) and potential (u) energy:

e = k + u

the total energy is e=675 j, and this does not change.

initially, the kinetic energy is k=296 j, so the potential energy is

at the end, the kinetic energy is k=432 j, so the potential energy is

so, we see that the potential energy has decreased from 379 j to 243 j.