Working with collinearity
When you're working in three dimensions, the only way to prove that three points are in a line (collinear) involves showing that a common direction exists. For this, you need to use vectors.
Here's how you would show that \(A(4,1,3)\), \(B(8,4,6)\) and \(C(20,13,15)\) are collinear.
First, choose two directed line segments with a common point:
\(\overrightarrow {AB} = \left( \begin{array}{l} 4\\ 3\\ 3 \end{array} \right),\,\overrightarrow {BC} = \left( \begin{array}{l} 12\\ \,\,9\\ \,\,9 \end{array} \right)\)
Express one as a multiple of the other:
\(\overrightarrow {BC} = 3\left( \begin{array}{l} 4\\ 3\\ 3 \end{array} \right)\), ie \(\overrightarrow {BC} = 3 \times \overrightarrow {AB}\)
and state a conclusion.
So \(\overrightarrow {AB}\) and \(\overrightarrow {BC}\) have a common direction.
Complete the proof.
\(\overrightarrow {AB}\) and \(\overrightarrow {BC}\) have a common point. Therefore \(A\), \(B\) and \(C\) are collinear.
Question
If \(\overrightarrow {PR} = \left( \begin{array}{l} \,\,\,\,\,5\\ - 1\\- 2 \end{array} \right),\,\overrightarrow {QR} = \left( \begin{array}{l} - 5\\ \,\,\,\,\,1\\ \,\,\,\,\,2 \end{array} \right)\) show that \(P\),\(Q\) and \(R\) are collinear.
\(\overrightarrow{QR}=- 1\left(\begin{array}{l}\,\,\,\,\,5\\- 1\\- 2\end{array}\right)\), ie \(\overrightarrow {QR} = - 1 \times \overrightarrow {PR}\)
So \(\overrightarrow {QR}\) and \(\overrightarrow {PR}\) have a common direction.
\(\overrightarrow {QR}\) and \(\overrightarrow {PR}\) have a common point \(R\).
Therefore \(P\), \(Q\) and \(R\) are collinear.
