Mon Prof Augmenté
Three points on one line
Age 15 · Maths
Open to everyone until Tuesday, November 17
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Two vectors are collinear when one is a number times the other, so they point along the same line. If the vector from A to C is a multiple of the vector from A to B, then A, B and C all lie on one straight line.
In the picture
- Starting point A: both vectors begin here
- Short vector AB: goes from A to B
- Long vector AC: same direction, exactly twice as long, so AC = 2·AB
- Point C: on line AB, because AC is a multiple of AB
Examples
- A drone flying at a steady speed in a fixed direction: its move after 2 minutes is twice its move after 1 minute, so its three positions are in a straight line.
- Three lampposts on a straight street: walking from the first to the third takes twice as many steps, in the same direction, as walking from the first to the second.
Key takeaways
- If AC = k·AB for some number k, then A, B and C are on one line.
- A negative k flips the direction, but the vectors are still collinear.
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