1.4.1Foundations in Physics
Scalars and Vectors
5 min read 10 parts Lesson 9 of 19 1 quiz Virtual lab
Lesson summary
1.4.1 Scalars and Vectors — AS PHYSICS 9702 notes: Tell scalars from vectors, add and subtract vectors graphically and by components, and resolve a vector into perpendicular components.Scalar = a quantity with magnitude (size) only.Vector = a quantity with both magnitude and dire…
- Scalars include
- Vectors include
- Work and energy are scalars
- Displacement
- An object can have constant speed but changing velocity
- A vector is drawn as an arrow
Section 1 of 10Scalars and vectors
Scalars and vectors
Scalar
Scalar a quantity with magnitude (size) only.
Vector
Vector a quantity with both magnitude and direction.
Scalars include
- Scalars include distance, speed, mass, time, energy, power, temperature, volume, density, pressure and electric charge.
Vectors include
- Vectors include displacement, velocity, acceleration, force, momentum and weight.
Work and energy are scalars
- Work and energy are scalars even though they are found from vectors.
Section 2 of 10Distance and displacement; speed and velocity
Distance and displacement; speed and velocity
Distance
- Distance the total length of the path travelled, with no direction, so it is a scalar.
Displacement
- Displacement the length and direction of the straight line from the start to the finish, so it is a vector.
Speed
- Speed distance per unit time, a scalar.
Velocity
- Velocity displacement per unit time, a vector.
An object can have constant speed but changing velocity
- An object can have constant speed but changing velocity when its direction changes.
Section 3 of 10Representing vectors
1A vector is drawn as an arrow
- A vector is drawn as an arrow the length is proportional to the magnitude and the arrowhead shows the direction.
2Vectors can be moved around a diagram
- Vectors can be moved around a diagram as long as their length and direction are unchanged.
3A negative vector
- A negative vector has the same magnitude but the opposite direction.
4Give direction as a compass bearing or an angle to a reference line
- Give direction as a compass bearing or an angle to a reference line for example 30° above the horizontal.
Section 4 of 10Adding vectors by the triangle method
1
Draw the first vector to scale.
2
Draw the second vector starting from the tip of the first (head to tail).
3
Draw the resultant from the tail of the first vector to the head of the last.
4
Measure the resultant and its angle, or calculate them.
Section 5 of 10Parallelogram method and calculation
Parallelogram method and calculation
Parallelogram method
- Parallelogram method link the vectors tail to tail, complete the parallelogram and take the diagonal as the resultant.
Resultant
Resultant the single vector that has the same effect as the vectors together.
Perpendicular vectors
- Perpendicular vectors use Pythagoras, R = √(A2 + B2), and tan θ = B ÷ A.
Other angles
- Other angles use the cosine and sine rules.
Adding vectors is not adding magnitudes
- 3 N and 4 N give between 1 N and 7 N, and exactly 5 N when perpendicular.
Section 6 of 10Subtracting vectors
Subtracting vectors
Subtracting a vector
- Subtracting a vector add the vector reversed, P − Q = P + (−Q).
Change in velocity
- Δv = vfinal − vinitial.
A ball bouncing off a wall at 5.0 m s−1 and returning at the same speed
- A ball bouncing off a wall at 5.0 m s−1 and returning at the same speed its speed is unchanged but its velocity changes by 10 m s-1.
Section 7 of 10Resolving a vector
1Resolving
- Resolving replacing a vector by two perpendicular components that together have the same effect.
2With θ measured from the horizontal
- With θ measured from the horizontal horizontal component Fx = F cos θ and vertical component Fy = F sin θ.
3If the angle is from the vertical
- If the angle is from the vertical cosine and sine swap.
4
Components along one axis do not influence motion along the perpendicular axis.
5Inclined plane
- Inclined plane weight resolves to W sin θ along the slope and W cos θ into it.
6To add several vectors
- To add several vectors resolve each, add x and y components separately, then recombine with Pythagoras and tangent.
Section 8 of 10Equilibrium and relative motion
Equilibrium and relative motion
Forces in equilibrium
- Forces in equilibrium the resultant force is zero and three coplanar forces form a closed vector triangle.
A hanging sign on two cables
- A hanging sign on two cables the vertical components of tension support the weight and the horizontal components cancel.
Relative velocity
- Relative velocity a boat's velocity relative to the bank is the vector sum of its velocity relative to the water and the water current.
Section 9 of 10Key terms
| Term | Meaning |
|---|---|
| Scalar | A quantity with magnitude only. |
| Vector | A quantity with magnitude and direction. |
| Resultant | The single vector with the same effect as several vectors. |
| Component | One of two perpendicular vectors that replace a single vector. |
| Resolving | Splitting a vector into perpendicular components. |
| Displacement | Distance from the start in a stated direction, a vector. |
| Velocity | Rate of change of displacement, a vector. |
| Closed vector triangle | A triangle of three forces that shows equilibrium. |
Section 10 of 10Exam technique
1
Adding magnitudes of vectors directly — use direction, so 3 N and 4 N do not always make 7 N.
2
Using cos for the vertical component when the angle is from the horizontal — the vertical component is F sin θ.
3
Treating distance and displacement, or speed and velocity, as the same.
4
Placing vectors head to head or tail to tail in a triangle — they must follow tip to tail.
5
Giving only a magnitude when the answer is a vector — state the direction too.