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.
More to read
Proving a Triangle Fact with Vectors
Draw a triangle with corners A, B, and C, and call its three sides vector A, vector B, and vector C.
Now mark the midpoint of side A and the midpoint of side B, and draw a new vector C' connecting those two midpoints.
You can prove that vector C' is always parallel to vector C, and exactly half its length, no matter what shape the triangle is. You do it the same way you prove any vector fact: write C' as a route using the two other side vectors, then show it comes out to be a plain number times C.
Once you see that C' is a fraction of C, like half of it, you know two things at once: the two vectors point along parallel lines, and one is exactly half the length of the other.
What Does 'Parallel' Really Mean?
Geometry has been asking questions about points, lines, and planes since Euclid wrote it all down around 300 BCE. One idea that keeps showing up is what it means for two lines to be parallel: they run in the same direction and never cross, no matter how far you extend them.
When you write directions as vectors instead of drawing them, 'same direction' turns into something you can check with numbers. If one vector is just a plain number times another vector, say 3 times as long but pointing the same way, then the lines they sit on are parallel. That's the vector version of the old geometry idea.
This trick also helps with points. If you can show that the route from one point to a second point is a plain number times the route from that same point to a third point, then all three points must sit on a single straight line, since there's only one direction available once two points and a shared starting point are fixed.
2 · Watch
En anglaisTake it off screen
Where it sits
Optional connections
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
17 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.