Chapter 10, Problem 40RE

To determine

Find whether given points are collinear using vectors.

Answer to Problem 40RE

The points $P\left(5,–4,7\right),Q\left(8,–5,5\right),R\left(11,6,3\right)$ are not collinear

Explanation of Solution

Given:

The points

  $P\left(5,–4,7\right),Q\left(8,–5,5\right),R\left(11,6,3\right)$

If there are two points in space $P\left(x_1,y_1,z_1\right)$ and $Q\left(x_2,y_2,z_2\right)$ then it can be written in component form as $\overline{PQ}=\langle \left(x_2-x_1\right),\left(y_2-y_1\right),\left(z_2-z_1\right)\rangle$

Three points are said to be collinear if the component vector $\overline{PQ}$ and $\overline{PR}$ of these points is in the form $\overline{PQ}=c\overline{PR}$

Consider the following points

  $P\left(5,–4,7\right),Q\left(8,–5,5\right),R\left(11,6,3\right)$

First compute the component forms of $\overline{PQ}$ and $\overline{PR}$ that is,

  $\begin{array}{l}\overline{PQ}=\langle 8-5,-5+4,5-7\rangle \hfill \\ =\langle 3,-1,-2\rangle \hfill \end{array}$

  $\begin{array}{l}\overline{PR}=\langle 11-5,6+4,3-7\rangle \hfill \\ =\langle 6,10,4\rangle \hfill \end{array}$

Since, $\overline{PQ}\ne c\overline{PR}$ where c is some constant.

Therefore, the points P(5,−4,7),Q(8,−5,5),R(11,6,3) are not collinear.