Vectors
Introduced| English |
|---|
| vector/ˈvektə/ |
| magnitude/ˈmæɡnɪtjuːd/ |
| scalar/ˈskeɪlə/ |
| component form/kəmˈpəʊnənt fɔːm/ |
Size and direction together
- "Drive $5$ km" is incomplete — you need to know which way.
- Many quantities are like this: velocity, force, displacement all carry a direction.
- A single number cannot capture both how much and which way.
- We need a new object — the vector — drawn as an arrow.
What a vector is
- A vector 向量 has two pieces of information: a magnitude 大小 (length) and a direction.
- It is drawn as an arrow: the length is the magnitude, the way it points is the direction.
- Contrast a scalar 标量 — a plain number with size only, like temperature or mass.
- Velocity is a vector; speed (just the number) is a scalar.
A vector is a quantity with…
A vector carries both size and direction, unlike a scalar (a plain number with size only).
Component form
- Break a vector into how far it goes across and up: its component form 分量形式.
- $\langle 3, 2\rangle$ means $3$ to the right and $2$ up.
- The horizontal and vertical components are like the legs of a right triangle.
- Adding vectors is as easy as adding their components separately.
Add two vectors tip to tail
Drag the two vectors and watch their sum. A vector has both a direction and a magnitude, unlike a plain number.
The component form $\langle 3, 2\rangle$ means the vector goes…
Component form lists the horizontal change then the vertical change: $\langle 3, 2\rangle$ is 3 right, 2 up.
What is the magnitude of the vector $\langle 3, 4\rangle$?
Magnitude $= \sqrt{3^2 + 4^2} = \sqrt{25} = 5$ — the length of the arrow, by the Pythagorean theorem.
Select all true statements about vectors.
A single number is a scalar, not a vector. The other three are correct.
Finding the magnitude
- The magnitude is the arrow's length, found by the Pythagorean theorem.
- For $\langle a, b\rangle$, the magnitude is $\sqrt{a^2 + b^2}$.
- So $\langle 3, 4\rangle$ has magnitude $\sqrt{9 + 16} = 5$.
- The components are the legs; the magnitude is the hypotenuse.
Multiplying a vector by the scalar $2$ doubles its length but keeps its direction.
A positive scalar scales the magnitude and leaves the direction unchanged; a negative one also reverses it.
Scaling and adding
- Multiply a vector by a scalar to stretch or shrink it: $2\langle 3, 2\rangle = \langle 6, 4\rangle$.
- A negative scalar also flips its direction.
- Add two vectors by adding components, or geometrically tip-to-tail.
- These operations let vectors model combined forces and motions.
Do not confuse a vector's magnitude with its components. The vector $\langle 3, 4\rangle$ has components $3$ and $4$, but magnitude $5$ — not $7$. The length is the hypotenuse, never the sum of the parts.
A boat heads $\langle 6, 8\rangle$ (km east, km north). How far has it travelled?
- The distance is the magnitude: $\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$ km.
- Its direction is up-and-to-the-right, steeper than $45°$ (more north than east).
- Doubling the vector to $\langle 12, 16\rangle$ would double the distance to $20$ km.
A vector carries both magnitude and direction, drawn as an arrow — unlike a scalar (a plain number). Its component form $\langle a, b\rangle$ lists horizontal and vertical parts, and its magnitude is $\sqrt{a^2 + b^2}$. Scale by multiplying, add by combining components.