1. I did the exercises last night because they were obligatory.
— I didn't have to do the exercises last night because they were obligatory.
Explanation: "didn't have to" is used to express that there was no obligation or necessity to do something. In this case, the person did the exercises voluntarily even though they were mandatory.
2. Taking food to the party was optional for us, but we took some.
— We needn't have taken food to the party, but we did.
Explanation: "needn't have" is used to express that something was not necessary in the past, but it was done anyway. In this case, the person took food to the party even though it was not necessary.
3. She didn't revise for the exam and that was a big mistake.
— She shouldn't have not revised for the exam.
Explanation: "shouldn't have" is used to express regret or disapproval for something that was done or not done in the past. In this case, the person is expressing disapproval for the fact that she didn't revise for the exam.
4. They didn't have permission to go into the office.
— They weren't allowed to go into the office.
Explanation: "weren't allowed to" is used to express that someone was prohibited or not given permission to do something. In this case, they didn't have the authority or consent to enter the office.
5. It was necessary for the plane to land.
— The plane had to land.
Explanation: "had to" is used to express that something was necessary or required to be done. In this case, the plane had no choice but to land.
6. It wasn't necessary to show my receipt to the shop assistant, so I didn't.
— I didn't need to show my receipt to the shop assistant.
Explanation: "didn't need to" is used to express that something was not necessary or required to be done. In this case, there was no need to show the receipt, so the person didn't do it.
7. I got angry with you yesterday and that was a silly thing to do.
— I ought not to have gotten angry with you yesterday.
Explanation: "ought not to have" is used to express regret or disapproval for something that was done in the past. In this case, the person is expressing regret for getting angry with the other person yesterday.
i*i = 1 (произведение основных векторов i*i равно 1)
i*2j = 2j (произведение основного вектора i на j равно 2j)
i*(-4k) = -4k (произведение основного вектора i на k равно -4k)
j*i = -j (произведение основного вектора j на i равно -j)
j*2j = 2j^2 (произведение основного вектора j на себя равно 2j в квадрате)
j*(-4k) = -4kj (произведение основного вектора j на k равно -4kj)
k*i = ki (произведение основного вектора k на i равно ki)
k*2j = 2kj (произведение основного вектора k на j равно 2kj)
k*(-4k) = -4k^2 (произведение основного вектора k на себя равно -4k в квадрате)
Теперь подставим найденные значения в формулу для скалярного произведения:
— I didn't have to do the exercises last night because they were obligatory.
Explanation: "didn't have to" is used to express that there was no obligation or necessity to do something. In this case, the person did the exercises voluntarily even though they were mandatory.
2. Taking food to the party was optional for us, but we took some.
— We needn't have taken food to the party, but we did.
Explanation: "needn't have" is used to express that something was not necessary in the past, but it was done anyway. In this case, the person took food to the party even though it was not necessary.
3. She didn't revise for the exam and that was a big mistake.
— She shouldn't have not revised for the exam.
Explanation: "shouldn't have" is used to express regret or disapproval for something that was done or not done in the past. In this case, the person is expressing disapproval for the fact that she didn't revise for the exam.
4. They didn't have permission to go into the office.
— They weren't allowed to go into the office.
Explanation: "weren't allowed to" is used to express that someone was prohibited or not given permission to do something. In this case, they didn't have the authority or consent to enter the office.
5. It was necessary for the plane to land.
— The plane had to land.
Explanation: "had to" is used to express that something was necessary or required to be done. In this case, the plane had no choice but to land.
6. It wasn't necessary to show my receipt to the shop assistant, so I didn't.
— I didn't need to show my receipt to the shop assistant.
Explanation: "didn't need to" is used to express that something was not necessary or required to be done. In this case, there was no need to show the receipt, so the person didn't do it.
7. I got angry with you yesterday and that was a silly thing to do.
— I ought not to have gotten angry with you yesterday.
Explanation: "ought not to have" is used to express regret or disapproval for something that was done in the past. In this case, the person is expressing regret for getting angry with the other person yesterday.
Итак, заданы векторы:
b = 4i - 3j + 5k
c = i + 2j - 4k
Чтобы найти скалярное произведение b·c, распишем его по формуле:
b·c = 4*i*i + 4*i*2j + 4*i*(-4k) - 3j*i - 3j*2j - 3j*(-4k) + 5k*i + 5k*2j + 5k*(-4k)
Теперь перемножим каждую пару коэффициентов:
i*i = 1 (произведение основных векторов i*i равно 1)
i*2j = 2j (произведение основного вектора i на j равно 2j)
i*(-4k) = -4k (произведение основного вектора i на k равно -4k)
j*i = -j (произведение основного вектора j на i равно -j)
j*2j = 2j^2 (произведение основного вектора j на себя равно 2j в квадрате)
j*(-4k) = -4kj (произведение основного вектора j на k равно -4kj)
k*i = ki (произведение основного вектора k на i равно ki)
k*2j = 2kj (произведение основного вектора k на j равно 2kj)
k*(-4k) = -4k^2 (произведение основного вектора k на себя равно -4k в квадрате)
Теперь подставим найденные значения в формулу для скалярного произведения:
b·c = 4*1 + 4*2j + 4*(-4k) - 3*(-j) - 3*2j^2 - 3*(-4kj) + 5ki + 5*2kj + 5*(-4k^2)
Упростим полученное выражение:
b·c = 4 + 8j - 16k + 3j - 6j^2 + 12kj + 5ki + 10kj - 20k^2
Объединим подобные члены:
b·c = 4 + 8j + 3j + 5ki + 12kj + 10kj - 16k - 20k^2 - 6j^2
b·c = 4 + 11j + 15ki + 22kj - 16k - 6j^2 - 20k^2
Таким образом, скалярное произведение векторов b и c равно 4 + 11j + 15ki + 22kj - 16k - 6j^2 - 20k^2.