Question 7
$A$, $B$ and $C$ are points in three-dimensional space.
If $2\overrightarrow{AB}=\overrightarrow{BC}$, then
Not quite. It’s the other way round. $\overrightarrow{BC}$ is twice $\overrightarrow{AB}$, so $|\overrightarrow{BC}|$ is twice $|\overrightarrow{AB}|$. In 2024, 25.8% of Queensland students chose (A).
Not quite. One vector is a multiple of the other, so they are parallel. Their dot product is $2|\overrightarrow{AB}|^2$, which isn’t zero. In 2024, 6.2% of Queensland students chose (B).
Not quite. Three points on one straight line lie in infinitely many planes, like the pages of a book all sharing the spine. Only one plane would contain them if they weren’t on a line. In 2024, 18.7% of Queensland students chose (C).
That’s it. Step through the hints below to see why. In 2024, 48.7% of Queensland students chose (D).
However you move the sliders, the three points stay on one straight line.
When one vector is a scalar multiple of another, the two vectors are parallel. Here $\overrightarrow{BC}$ points the same way as $\overrightarrow{AB}$ and is twice as long.
The two vectors are parallel and they share the point $B$. Parallel lines through a common point are the same line, so $A$, $B$ and $C$ are collinear.
$$\overrightarrow{AC}=\overrightarrow{AB}+\overrightarrow{BC}=3\overrightarrow{AB}$$
Rule out the others. The lengths are the wrong way round in (A). Parallel vectors aren’t perpendicular, so (B) is out. A line lies in infinitely many planes, so (C) is out too.
$\overrightarrow{BC}$ is a scalar multiple of $\overrightarrow{AB}$, so the vectors are parallel. They share the point $B$, so $A$, $B$ and $C$ lie on one straight line.
How Queensland students answered in 2024
From the QCAA Specialist Mathematics subject report for 2024. The percentages can add to less than 100 because some students didn’t answer.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording is from the 2024 QCAA Specialist Mathematics external assessment, © State of Queensland (QCAA) 2024, licensed under CC BY 4.0, and has been adapted. Tangent Tuition is not affiliated with the QCAA.
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