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 $\vec{AB}$ (b) Show that quadrilateral OABC is a trapezium, giving reasons for your answer. - Edexcel - A-Level Maths Pure - Question 5 - 2020 - Paper 1

To determine if quadrilateral OABC is a trapezium, we need to demonstrate that one pair of opposing sides is parallel.

We already found $\vec{AB} = 1i - 8j + 2k$.

Next, we find the vector $\vec{OC}$:

$\vec{C} = 2i - 16j + 4k$
$\vec{OC} = \vec{C} - \vec{O} = 2i - 16j + 4k$ (since O is the origin, its position vector is $0$)

Now, we’ll compare this with the vector representation of line segment $\vec{AB}$:

From the calculations of $\vec{AB}$, we can establish a relationship:

$\vec{OC} = 2(1i - 8j + 2k) = 2\vec{AB}$

Since $\vec{OC}$ is a scalar multiple of $\vec{AB}$, they are parallel. Therefore, with one pair of opposite sides (AB and OC) being parallel, we conclude that quadrilateral OABC is a trapezium.