Question

question
consider three points on a plane, ( s = (-3,87) ), ( t = (3,89) ), and ( u = (6,90) ).
use your equation to show that ( u ) is on the same line as ( s ) and ( t ).
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Explanation:

Step1: Find the vector \(\overrightarrow{ST}\)

The formula for the vector between two points \(S(x_1,y_1)\) and \(T(x_2,y_2)\) is \(\overrightarrow{ST}=(x_2 - x_1,y_2 - y_1)\).
Given \(S=(-3,87)\) and \(T=(3,89)\), then \(\overrightarrow{ST}=(3-(-3),89 - 87)=(6,2)\)

Step2: Find the vector \(\overrightarrow{TU}\)

Given \(T=(3,89)\) and \(U=(6,90)\), then \(\overrightarrow{TU}=(6 - 3,90 - 89)=(3,1)\)

Step3: Check the scalar - multiple relationship

Notice that \(\overrightarrow{ST}=(6,2)=2\times(3,1) = 2\overrightarrow{TU}\)

Since \(\overrightarrow{ST}\) is a scalar multiple of \(\overrightarrow{TU}\), the points \(S\), \(T\), and \(U\) are collinear. That is, \(U\) is on the same line as \(S\) and \(T\)

Answer:

Since \(\overrightarrow{ST}=(6,2)\) and \(\overrightarrow{TU}=(3,1)\) with \(\overrightarrow{ST} = 2\overrightarrow{TU}\), the points \(S\), \(T\), and \(U\) are collinear. So \(U\) is on the same line as \(S\) and \(T\)