A vector = how far AND which way: Picture a drone leaving its base. Saying it flew 5 km is not enough — you need a direction too. A vector stores both: a size (magnitude) and a direction.
We record it in components — how far across (x) and how far up (y):
means 3 across, 4 up.
A plain number with no direction (like the air temperature, or a price) is called a scalar.
Why the square-root formula works: The components are the two legs of a right-angled triangle, and the vector itself is the hypotenuse.
Pythagoras gives the hypotenuse, so the length is . In 3D you just add the third component squared.
IB-style question — magnitude of a displacement
A drone's displacement from base is km (east, north).
How far is the drone from base?
Step by step
- The distance from base is the magnitude of the vector — use Pythagoras on the components.
- Substitute the components 6 and 8.
- Take the square root.
Final answer
The drone is 10 km from base. (The components give the direction; the magnitude gives the straight-line distance.)
IB-style question — magnitude in 3D
A delivery drone rises while it flies, with displacement (in km).
Find .
Step by step
- In 3D, add the third component squared as well.
- Substitute 2, 3 and 6.
- Take the square root.
Final answer
The drone is 7 km from base in a straight line.
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Combine vectors component by component: To add two vectors, add their matching components (do the first journey, then the second — 'tip to tail').
To subtract, subtract matching components. To scalar-multiply by a number , multiply every component by — this stretches the vector (and flips it if is negative). Its direction stays the same line unless the sign flips.
IB-style question — combine two legs of a route
A delivery van drives km, then km.
Find the total displacement and the difference .
Step by step
- Add matching components for the total journey.
- Subtract matching components for the difference.
Final answer
Total displacement km; difference km.
IB-style question — scaling a force
A small motor pushes with force N. An identical motor is added, then the direction is reversed.
Write and .
Step by step
- Two motors double every component (scalar multiply by 2).
- Reversing direction multiplies by −1 (same size, opposite way).
Final answer
N (twice as strong, same direction); N (same size, opposite direction).
Unit vector — point the same way, length 1: A unit vector has length exactly 1. To build one, take any vector and divide it by its own magnitude:
The little hat means 'length one'. It keeps the direction but throws away the size — useful when you only care which way something points.
IB-style question — unit vector of a current
A sea current flows along m s⁻¹.
Find the unit vector in the direction of the current.
Step by step
- First find the magnitude (how fast the current flows).
- Divide each component by the magnitude.
- Check: a unit vector must have length 1.
Final answer
— it points the same way as the current but has length 1.