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Scalar and Vector Quantities is taught in the note for Applications of Vectors. Read it below and look for the section on this topic.

Applications of Vectors

12 min read

Scalar and vector quantities

A scalar has only a size (magnitude): mass, time, distance, speed, energy, temperature. A vector has a size and a direction: displacement, velocity, acceleration, force, momentum. Vectors are drawn as arrows. The length of the arrow shows the size and the arrowhead shows the direction.

ScalarVector
DistanceDisplacement
SpeedVelocity
MassWeight (a force)
Energy, work, powerForce, acceleration, momentum

Adding vectors

To add vectors, draw them head to tail. The resultant is the single vector from the start of the first to the end of the last. For two vectors at right angles, use Pythagoras: R = √(A² + B²). For any angle θ between them, use the cosine rule.

Adding vectorsTwo vectors drawn head to tail. Vector A is 4 units to the right and vector B is 3 units upwards. The resultant is the straight line from the start of A to the end of B, which is 5 units long. A parallelogram gives the same result.A = 4B = 3resultant RR = √(A² + B²) = √(16 + 9) = 5parallelogram method
θ is the angle between the two vectors when they are drawn from the same point. At θ = 90° this becomes R = √(A² + B²); at 0° the vectors add (R = A + B); at 180° they subtract.
Try itResultant of two vectors

Choose two forces and the angle between them. See the size of the resultant.

Resolution of vectors

Resolving a vector means replacing it by two perpendicular components that together have the same effect. A vector of size F at angle θ to the horizontal has a horizontal component F cos θ and a vertical component F sin θ.

Resolving a vector into componentsA force of 10 newtons acts at 30 degrees above the horizontal. It can be replaced by a horizontal component of 10 cosine 30, which is 8.7 newtons, and a vertical component of 10 sine 30, which is 5 newtons. The two components together have the same effect as the original force.θ = 30°F = 10 NF cos θ = 8.7 NF sin θ = 5 N
Worked examplePulling a trolley

A rope pulls a trolley with a force of 50 N at 40° above the horizontal. Find the force that moves it forward and the force that lifts it.

  1. Horizontal component.
  2. Vertical component.

Answer: The pulling force forward is 38.3 N and the lifting force is 32.1 N.

Relative motion

The velocity of A relative to B is the velocity of A as seen by B: v(AB) = v(A) − v(B) (vector subtraction). Two cars moving the same way at 80 km/h and 60 km/h are separated at 20 km/h; moving towards each other they approach at 140 km/h. A boat crossing a river has its own velocity (through the water) plus the velocity of the river, so it is carried downstream.

Check yourself Which of these is a vector?
Check yourself Two forces of 6 N and 8 N act at right angles. The resultant is…
Check yourself A force of 100 N acts at 60° to the horizontal. Its horizontal component is…

Key terms

Scalar
A quantity with size only.
Vector
A quantity with size and direction.
Resultant
The single vector with the same effect as several vectors.
Component
One of two perpendicular vectors that together replace a vector.

Summary

  • Scalars have size; vectors have size and direction.
  • Add vectors head to tail; for perpendicular vectors R = √(A² + B²).
  • A vector F at angle θ has components F cos θ and F sin θ.
  • Relative velocity is found by vector subtraction.
Relative Motion