Chapter 10

Vectors in Two Dimensions

Vectors as displacements rather than points, adding them by following one after another, and the relative-velocity problems that are the reason this topic is on the syllabus.

หลักสูตรแกนกลาง / สสวท.: การวัดและเรขาคณิต (Measurement and Geometry)ตรงกับ ม.5 เวกเตอร์ในสามมิติ (เพิ่มเติม) แต่ 0606 อยู่ในระนาบสองมิติเท่านั้น
Common Core (US): N-VM
Cambridge: IGCSE 0606 — Topic 13: Vectors in two dimensions

10.1 Notation, magnitude and direction

นิยาม

Vector

A quantity with both size and direction. It says how far and which way to MOVE, not where you are — which is why the same vector can be drawn anywhere on the page and still be the same vector.

Magnitude and unit vector

a = (x y) ∣ a ∣ = x^{2} + y^{2} \hat{a} = \frac{a}{∣ a ∣}

A unit vector has magnitude $1$ and keeps the direction. Dividing by the magnitude is what strips out the size and leaves the direction alone.

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Rules that follow straight from "a vector is a movement"
- $A B = - B A$ — the same journey the other way
- $A B + B C = A C$ — go via $B$ or go direct, you end up in the same place
- $A B = b - a$ when the position vectors are $a$ and $b$
- if $p = k q$ then the two are parallel

ข้อควรระวัง

$A B = b - a$ — the finish minus the start, which feels backwards the first ten times. Getting it the wrong way round produces a vector of the right length pointing in exactly the wrong direction, and nothing in the arithmetic afterwards complains.

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