Proving Geometry Facts with Vectors
Show that one vector expression is a multiple of another, and use that to prove lines are parallel or that three points lie on a straight line.
What a learner can do afterwards
- Write a route in terms of the two given vectors before comparing anything
- Argue that one vector is a multiple of another, so the lines are parallel
- Use a shared point plus parallel vectors to show three points are in a line
1 · Read
Write every route with the vectors you know. From A to B, undo the trip to A, then do the trip to B. With OA as a and OB as b, vector AB is b minus a. The midpoint M of AB sits at half of a plus b from O.
One vector that is a plain number times another points the same way. PQ of 2a plus 6b is exactly 2 times RS of a plus 3b, so the lines are parallel. Match the a parts and the b parts to find the number: 6a minus 3b is 1.5 times 4a minus 2b.
Parallel alone is not enough for one straight line. Add a shared point: equal vectors through Q lock P, Q and R onto a single line. State the shared point in your proof, or the argument falls short.
The midpoint connector proof ties it together. In a triangle, the route joining two midpoints works out to half of the third side vector. Half the length and the same direction means parallel, which is exactly what was claimed.
Write routes with known vectors, test for a number multiple, and add a shared point for collinearity.
2 · Watch
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.