A vector is a journey, not a place: Imagine telling a friend how to get from one spot to another: "go 3 steps right and 4 steps up." That instruction is a vector — it has a direction and a size (length), but no fixed starting point. Slide it anywhere and it's still the same vector.
We write that journey two equivalent ways:
Column form: stack the components — across on top, up underneath.
Base-vector form: i means "1 step in the x-direction", j means "1 step in the y-direction" (and in 3D, k means "1 step in the z-direction").
IB-style question — switch between notations
A drone's displacement is given as 5i − 2j + 6k metres.
Write this displacement in column form, and state each component.
Step by step
- The number in front of i is the x-component, in front of j is the y-component, in front of k is the z-component. Just read them off and stack them.
- Each entry is a signed step: positive = forward/right/up, negative = the opposite way.
Final answer
Column form (5, −2, 6)ᵀ; components x = 5, y = −2, z = 6 (metres).
Length = Pythagoras on the components: A vector's magnitude (written |v|) is just how long the arrow is. Picture the 2D vector (3, 4) as the hypotenuse of a right triangle with legs 3 and 4 — Pythagoras gives length √(3² + 4²) = √25 = 5.
The same idea works in 3D: square every component, add them, square-root. The magnitude is always ≥ 0 (a length can't be negative).
IB-style question — 2D magnitude
Find the magnitude of the vector a = 6i − 8j.
Step by step
- Write the magnitude formula with the two components.
- Substitute x = 6, y = −8. Squaring removes the sign, so −8 becomes 64.
- Add and square-root.
Final answer
|a| = 10.
IB-style question — 3D magnitude
Find the magnitude of v = (2, −3, 6)ᵀ. Give an exact value.
Step by step
- Use the 3D length formula.
- Substitute the three components (signs disappear when squared).
- Add inside the root.
Final answer
|v| = 7.