01 · Journey
A walker goes 9 m east then 3 m west. Find total distance and displacement.
Hint
Add lengths for distance and directed changes for displacement.
Worked solution
Distance=9+3=12 m. Displacement=6 m east.
Understand · explore · practise
Scalars and vectors in mechanics
Distinguish scalar quantities from vectors, use a clear reference direction and avoid classifying quantities by positivity alone.
Before you startSI units and basic directions.
01 / Direction is part of a vector quantity
A sign alone does not decide the classification.
Scalars do not require spatial direction; vectors do.
Speed, distance, mass and elapsed duration are scalars. Velocity, displacement, acceleration and force are vectors. Zero values are allowed, and some scalar quantities can be negative relative to a chosen scale.
What information defines the quantity?Explore
Reveal the classification
02 / Separate path length from change of position
Similar units do not make two quantities identical.
02 · Return
A walker returns to the starting point after travelling 20 m in total. State distance and displacement magnitude.
Hint
Net change of position is zero.
Worked solution
Distance=20 m; displacement magnitude=0 m. A zero displacement does not imply no movement.
03 · Units
Why does the shared unit metre not tell you whether a quantity is distance or displacement?
Hint
The definitions contain different information.
Worked solution
Distance is a path length; displacement is a directed change of position. Units alone do not encode that distinction.
03 / Speed is the magnitude of velocity
A change of direction can change velocity without changing speed.
Watch: velocity turns while its magnitude stays fixed
Pause, replay or seek freely. The notes explain the same idea and stay in view.
04 · Opposite motion
Two cyclists each travel at 5 m/s, one east and one west. Are their speeds equal? Are their velocities equal?
Hint
Compare magnitude and direction separately.
Worked solution
Their speeds are equal. Their velocities are different because the directions differ.
05 · Signed velocity
With east positive, a velocity component is −3 m/s. State the speed and direction.
Hint
Take the magnitude for speed.
Worked solution
Speed=3 m/s, travelling west. The minus sign refers to the chosen axis.
06 · Rest
At an instant a particle has zero velocity. What is its speed? Must its acceleration be zero?
Hint
A turning point can have zero velocity.
Worked solution
Its speed is zero. Its acceleration need not be zero; for example a vertically thrown body has zero instantaneous velocity at its highest point but still accelerates downwards.
04 / Choose the reference direction before using signs
The same physical vector has different components on reversed axes.
A trolley travels west at 4 m/s.Worked example
East-positive axis: v=−4 m/s
West is the negative direction on this axis.
West-positive axis: v=+4 m/s
The physical velocity is unchanged; only its component description changes.
Speed is 4 m/s in both descriptions
A magnitude does not acquire the direction sign.
07 · Axis reversal
A force has component +7 N when right is positive. What is its component when left is positive?
Hint
Reverse the coordinate direction, not the physical force.
Worked solution
−7 N. The force still acts to the right.
08 · Position versus velocity
A particle is at x=−2 m. Can you determine its velocity direction from that fact alone?
Hint
Position and change of position are different.
Worked solution
No. It is left of the origin on a right-positive axis, but it could be moving either way or be instantaneously at rest.
05 / Zero does not break the classification
Direction for a zero vector need not be specified.
09 · Zero force
Is a zero resultant force still a vector quantity?
Hint
The type of quantity is defined by its role, not whether this value is zero.
Worked solution
Yes. It is the zero vector. It has zero magnitude and no unique direction.
10 · Zero scalar
Can a speed or an elapsed duration be zero?
Hint
Consider rest or the starting instant.
Worked solution
Yes. Speed can be zero at rest, and elapsed duration is zero at the chosen start. “Scalar” does not mean strictly positive.
06 / Negativity is not a vector test
A scalar scale can also contain negative values.
11 · Temperature
A temperature is −5°C. Does the minus sign make temperature a spatial vector?
Hint
Does it specify a spatial direction?
Worked solution
No. Temperature is a scalar; the negative value is relative to the Celsius scale, not a spatial direction.
12 · Classification
Classify mass, acceleration and force as scalar or vector.
Hint
Which require spatial direction?
Worked solution
Mass is scalar. Acceleration and force are vectors. Their signed components in a one-dimensional model refer to a chosen direction.
07 / Write enough information to remove ambiguity
A magnitude may not fully specify a vector.
13 · Force description
Why is “a force of 10 N” incomplete if the task asks for the full force vector?
Hint
Magnitude is only part of the information.
Worked solution
Its direction must also be stated, unless already fixed by the context or coordinate convention.
14 · Negative acceleration
Does acceleration −2 m/s² necessarily mean the particle is slowing down?
Hint
Compare acceleration with velocity direction.
Worked solution
No. It means acceleration in the negative direction. If velocity is also negative, the particle can speed up; if velocity is positive, it slows locally. Signs alone need context.
08 / Classify the physical quantity, then describe its value
State units and direction conventions.
Scalars do not carry spatial direction. Vectors do, and one-dimensional signs encode that direction relative to an axis. Speed is a magnitude; velocity is directed. Both scalars and vectors can be zero, and a negative number by itself does not identify a vector.
Section 1 of 8 · Direction is part of a vector quantity