1.4 Scalars and vectors
- Syllabus
- 9702–2028–2029
- Topic
- 1.4
- Level
- AS
Learning objectives
1.4.1The difference between scalar and vector quantities and give examples of• understand the difference between scalar and vector quantities and give examples of scalar and vector quantities included in the syllabus1.4.2Add and subtract coplanar vectors• add and subtract coplanar vectors1.4.3A vector as two perpendicular components• represent a vector as two perpendicular components
Scalars have magnitude; vectors have magnitude and direction
A scalar quantity is fully described by magnitude and unit. A vector quantity also requires direction; changing the direction changes the vector even when its magnitude is unchanged.
Scalars include distance, speed, mass, time, temperature and energy. Vectors include displacement, velocity, acceleration, force, weight and momentum.
A car travelling 5.0 m s⁻¹ east has speed 5.0 m s⁻¹ but velocity 5.0 m s⁻¹ east. The distance travelled is scalar; displacement from the starting point is vector.
A minus sign may encode direction along a chosen axis, but it does not by itself make a quantity a vector. The physical definition of the quantity determines whether direction is required.
Add vectors head-to-tail; subtract by reversing the vector
For head-to-tail addition, move a vector parallel to itself so its tail meets the previous head. The resultant points from the first tail to the final head. Translation does not change a vector's magnitude or direction.
To calculate A − B, reverse B to make −B, then add A + (−B). In components, add or subtract corresponding x- and y-components with their signs.
A force of 3.0 N east plus 4.0 N north has components (3.0, 4.0) N. Its resultant magnitude is 5.0 N and its direction is tan⁻¹(4/3) = 53° north of east.
Do not add magnitudes unless the vectors act along the same direction. Opposite directions require signs; other angles require geometry or components.
Resolve a vector along two chosen perpendicular axes
Two perpendicular components add vectorially to reproduce the original vector. Choose and label the axes first; each component is the projection of the vector onto one axis.
For a vector of magnitude V at angle θ measured from the positive x-axis: Vx = V cos θ and Vy = V sin θ. Give each component the sign set by its direction along the chosen axes.
A 10.0 N force at 30° above the horizontal has Fx = 10.0 cos 30° = 8.66 N horizontally and Fy = 10.0 sin 30° = 5.00 N upward.
Cosine gives the component adjacent to the stated angle; it is not automatically the horizontal component. If θ is measured from the vertical, the vertical component is V cos θ.