A vector represents a quantity with both magnitude (size) and direction, unlike a scalar, which has magnitude only. Vectors can be represented as column vectors (like (3, 4)) or with magnitude-direction notation, and their magnitude is found using Pythagoras' theorem: |v| = √(x² + y²). Vectors are added by combining their components, which is useful for modelling combined forces, velocities, or displacements in physics and engineering.
Example
A boat travelling at velocity (4, 0) km/h (east) in a current flowing at (0, 3) km/h (north) has a combined velocity vector of (4, 3), with a resulting speed (magnitude) of √(4² + 3²) = 5 km/h in a direction between east and north.
Key terms
Vector:
A quantity with both magnitude and direction.
Scalar:
A quantity with magnitude only, no direction.
Magnitude:
A vector's size or length, found using Pythagoras' theorem.
Questions
1. A vector has:
Both magnitude and direction
Magnitude only, with no direction
Direction only, with no magnitude
Neither magnitude nor direction
2. A scalar has:
Magnitude only
Both magnitude and direction
Direction only
Neither magnitude nor direction
3. A vector's magnitude is found using:
Pythagoras' theorem
Simple addition only
A completely unrelated formula
Only subtraction
4. A column vector might be written as:
(3, 4)
A single number only
A word with no numbers
A colour
5. Vectors are added by:
Combining their components
Ignoring one of the vectors completely
Only comparing their magnitudes
Never being combined at all
6. Vectors are useful for modelling:
Combined forces, velocities or displacements
Only single, unchanging numbers
Nothing physical or measurable
Only colours
7. A boat's speed (ignoring direction) would be considered:
A scalar quantity
Always a vector, never a scalar
Neither a vector nor a scalar
A quantity with no numerical value
8. The magnitude of vector (3, 4) equals:
5
7
12
1
9. The magnitude of vector (6, 8) equals:
10
14
48
2
10. Adding vectors (4, 0) and (0, 3) gives:
(4, 3)
(0, 0)
(4, 0)
(3, 4)
11. The magnitude of the combined vector (4, 3) equals:
5
7
12
1
12. Why is a vector's direction just as important as its magnitude when modelling real-world quantities like force?
The same magnitude of force applied in different directions produces very different physical effects
Direction never has any meaningful effect on a physical outcome
Magnitude alone always fully describes a force's effect
Force can always be fully understood using only a scalar quantity
13. Why might a plane's ground speed differ from its air speed when flying into a headwind?
The wind vector combines with (opposes) the plane's velocity vector, changing its resulting velocity relative to the ground
Wind has no effect on a plane's velocity relative to the ground
Ground speed and air speed are always exactly identical regardless of wind
Velocity vectors can never be combined with wind vectors in any scenario
14. Why is Pythagoras' theorem an appropriate tool for finding a vector's magnitude from its components?
A vector's horizontal and vertical components form a right-angled triangle with the vector itself as the hypotenuse
Pythagoras' theorem has no mathematical connection to vector components
Vector magnitude can only ever be measured directly, never calculated
The components of a vector never form any kind of triangle
15. A swimmer swimming directly across a river at (0, 2) m/s, in a current flowing at (3, 0) m/s, has a resulting velocity magnitude of:
√13
5
3
2
16. Why might engineers use vector addition to determine the net force acting on a structure from multiple different directions?
Combining all individual force vectors gives the single resulting force the structure actually experiences
Forces from different directions can never be mathematically combined
Only one force is ever considered when analysing a structure, regardless of direction
Net force calculations never require considering the direction of individual forces
17. Why might representing displacement as a vector (rather than just distance as a scalar) matter for navigation?
Direction is essential for determining where you actually end up, not just how far you travelled
Distance alone always fully describes where a journey ends up
Displacement and distance are always exactly identical quantities
Direction has no relevance to determining a final position
18. Subtracting vector (5, 2) from vector (8, 6) gives:
(3, 4)
(13, 8)
(3, -4)
(-3, 4)
19. Why might two vectors with the same magnitude but different directions represent very different physical quantities?
Direction determines the actual effect or outcome, so identical magnitude alone doesn't mean identical results
Vectors with identical magnitude always represent exactly the same physical quantity regardless of direction
Direction has no bearing on how a vector quantity behaves physically
Magnitude is the only property that matters when comparing two vectors
20. Why might a tug-of-war between two teams pulling with equal magnitude but opposite direction result in no net movement?
The two force vectors cancel out when added, since their components are equal and opposite
Equal and opposite forces always combine to produce a large net movement
Vector addition can never result in a resultant vector of zero magnitude
Direction has no role in determining the combined effect of two forces
21. Why might a hiker's total displacement differ from the total distance they walked if their path wasn't a straight line?
Displacement measures the straight-line vector from start to end, while distance measures the total path length actually travelled
Displacement and distance are always exactly the same value regardless of the path taken
A curved or indirect path never affects the relationship between distance and displacement
Distance is always smaller than displacement for any given path
Answer key (parent copy)
1. Both magnitude and direction
2. Magnitude only
3. Pythagoras' theorem
4. (3, 4)
5. Combining their components
6. Combined forces, velocities or displacements
7. A scalar quantity
8. 5
9. 10
10. (4, 3)
11. 5
12. The same magnitude of force applied in different directions produces very different physical effects
13. The wind vector combines with (opposes) the plane's velocity vector, changing its resulting velocity relative to the ground
14. A vector's horizontal and vertical components form a right-angled triangle with the vector itself as the hypotenuse
15. √13
16. Combining all individual force vectors gives the single resulting force the structure actually experiences
17. Direction is essential for determining where you actually end up, not just how far you travelled
18. (3, 4)
19. Direction determines the actual effect or outcome, so identical magnitude alone doesn't mean identical results
20. The two force vectors cancel out when added, since their components are equal and opposite
21. Displacement measures the straight-line vector from start to end, while distance measures the total path length actually travelled