Question 18(a): The diagram shows a parallelogram CDEF. $\overrightarrow{FC}=m$ and $\overrightarrow{CE}=n$. A is the midpoint of CD and B is the midpoint of CE. Find an expression, in terms of m and n, for $\overrightarrow{AB}$. Give your answer in its simplest form.

Question

Вопрос: $Фото задания: Question 18(a): The diagram shows a parallelogram CDEF. $\overrightarrow{FC}=m$ and $\overrightarrow{CE}=n$. A is the…$ Фото задания

Question 18(a): The diagram shows a parallelogram CDEF. $\overrightarrow{FC}=m$ and $\overrightarrow{CE}=n$. A is the midpoint of CD and B is the midpoint of CE. Find an expression, in terms of m and n, for $\overrightarrow{AB}$. Give your answer in its simplest form.

Accepted Answer

Перевод задания: На рисунке изображён параллелограмм CDEF. $\overrightarrow{FC}=m$, а $\overrightarrow{CE}=n$. A — середина CD, B — середина CE. Найдите выражение для $\overrightarrow{AB}$ через m и n и упростите ответ.

Let C be the origin. Then the position vectors of F and E are $-m$ and $n$, respectively. Since CDEF is a parallelogram, $\overrightarrow{CD}=\overrightarrow{FE}=m+n$, so the position vector of D is $m+n$.

As A and B are midpoints, their position vectors are $\frac{m+n}{2}$ and $\frac{n}{2}$, respectively. Therefore, $\overrightarrow{AB}=\frac{n}{2}-\frac{m+n}{2}=-\frac{1}{2}m$.

Пусть C — начало отсчёта. Тогда радиус-векторы точек F и E равны $-m$ и $n$. Поскольку CDEF — параллелограмм, $\overrightarrow{CD}=\overrightarrow{FE}=m+n$, поэтому радиус-вектор D равен $m+n$.

Так как A и B — середины отрезков, их радиус-векторы равны $\frac{m+n}{2}$ и $\frac{n}{2}$. Следовательно, $\overrightarrow{AB}=\frac{n}{2}-\frac{m+n}{2}=-\frac{1}{2}m$.

Answer: $\overrightarrow{AB}=-\frac{1}{2}m$.

Question 18(a): The diagram shows a parallelogram CDEF. \(\overrightarrow{FC}=m\) and \(\overrightarrow{CE}=n\). A is the midpoint of CD and B is the midpoint of CE. Find an expression, in terms of m and n, for \(\overrightarrow{AB}\). Give your answer in its simplest form.

Ответ: