Relative to a fixed origin O - point A has position vector $2i + 5j - 6k$ - point B has position vector $3i - 3j - 4k$ - point C has position vector $2i - 16j + 4k$ (a) Find $\overrightarrow{AB}$ (b) Show that quadrilateral OABC is a trapezium, giving reasons for your answer. - Edexcel - A-Level Maths Pure - Question 5 - 2020 - Paper 1

To show that quadrilateral OABC is a trapezium, we need to demonstrate that at least one pair of its opposite sides is parallel.

First, we consider the position vector for point C:

$\overrightarrow{C} = 2i - 16j + 4k$

Next, we need to compare the position vector $\overrightarrow{OC}$ with the vector $\overrightarrow{AB}$:

1. From our previous calculation, we have $\overrightarrow{AB} = i - 8j + 2k$
   and $\overrightarrow{OC} = 2i - 16j + 4k$

2. Now, we can observe that:

$\overrightarrow{OC} = 2 \cdot \overrightarrow{AB}$

Since $\overrightarrow{OC}$ is a scalar multiple of $\overrightarrow{AB}$, this implies that the two vectors are parallel.

Thus, since $\overrightarrow{OC}$ is parallel to $\overrightarrow{AB}$, it follows that quadrilateral OABC is a trapezium.