Working With Collinearity - Geometric Vectors - Higher Maths Revision

In this guide

1. Revise
2. Test

Pages

1. Geometric vectors
2. Determining the resultant in three dimensions
3. Working with collinearity

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.

Show answer

\(\overrightarrow{QR}=- 1\left(\begin{array}{l}\,\,\,\,\,5\\- 1\\- 2\end{array}\right)\), 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.

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• Working with vectors

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