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Worked solution

Calculate displacements and recognise a parallelogram

The position vectors of A,B,C,DA,B,C,D are (−14−3)\begin{pmatrix}-1\\4\\-3\end{pmatrix}, (15−2)\begin{pmatrix}1\\5\\-2\end{pmatrix}, (570)\begin{pmatrix}5\\7\\0\end{pmatrix} and (211)\begin{pmatrix}2\\1\\1\end{pmatrix} respectively. Evaluate (i) AB→\overrightarrow{AB}, (ii) BC→\overrightarrow{BC}, (iii) 2AB→2\overrightarrow{AB}, (iv) DB→\overrightarrow{DB}. Hence or otherwise determine (v) whether A,B,C are collinear and (vi) the shape of O,A,B,D.

CJC · Vectors Basic · Self-Practice 1

(i) AB

Answer

(211)\begin{pmatrix}2\\1\\1\end{pmatrix}.

Subtract A from B

AB→=(15−2)−(−14−3)=(211)\overrightarrow{AB}=\begin{pmatrix}1\\5\\-2\end{pmatrix}-\begin{pmatrix}-1\\4\\-3\end{pmatrix}=\begin{pmatrix}2\\1\\1\end{pmatrix}

These working steps are authored; the source prints the final answers.

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(ii) BC

Answer

(422)\begin{pmatrix}4\\2\\2\end{pmatrix}.

Subtract B from C

BC→=(570)−(15−2)=(422)\overrightarrow{BC}=\begin{pmatrix}5\\7\\0\end{pmatrix}-\begin{pmatrix}1\\5\\-2\end{pmatrix}=\begin{pmatrix}4\\2\\2\end{pmatrix}

Use terminal minus initial, including the signs of the coordinates.

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(iii) Twice AB

Answer

(422)\begin{pmatrix}4\\2\\2\end{pmatrix}.

Scale every coordinate

2AB→=2(211)=(422)2\overrightarrow{AB}=2\begin{pmatrix}2\\1\\1\end{pmatrix}=\begin{pmatrix}4\\2\\2\end{pmatrix}

This matches BC, which will establish collinearity.

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(iv) DB

Answer

(−14−3)\begin{pmatrix}-1\\4\\-3\end{pmatrix}.

Subtract D from B

DB→=(15−2)−(211)=(−14−3)\overrightarrow{DB}=\begin{pmatrix}1\\5\\-2\end{pmatrix}-\begin{pmatrix}2\\1\\1\end{pmatrix}=\begin{pmatrix}-1\\4\\-3\end{pmatrix}

DB starts at D, so the subtraction order is B minus D.

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(v) Collinearity

Answer

Yes: A,B,C are collinear.

Compare adjacent displacements

BC→=2AB→\overrightarrow{BC}=2\overrightarrow{AB}

AB and BC are nonzero and parallel, and share B.

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(vi) Shape of O,A,B,D

Answer

OABD is a parallelogram.

Match directed opposite sides

OA→=DB→=(−14−3),AB→=OD→=(211)\overrightarrow{OA}=\overrightarrow{DB}=\begin{pmatrix}-1\\4\\-3\end{pmatrix},\qquad\overrightarrow{AB}=\overrightarrow{OD}=\begin{pmatrix}2\\1\\1\end{pmatrix}

OA=DB and AB=OD. With the vertices in order O,A,B,D, both pairs of opposite sides match. The directions OA and OD are not parallel, so the shape is nondegenerate.

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Solution notes

All six printed answers are retained. The full explanations are authored supplements to the short answer key.