Scalar & Vector Quantities (College Board AP® Physics 1: Algebra-Based)

Scalar & vector quantities

• Some physical quantities in math and physics are scalar quantities and some are vector quantities

• Scalar quantities have only magnitude (no direction)

• Vector quantities have both magnitude and direction

  • For example, 10 m/s is a scalar quantity, and 10 m/s north is a vector quantity

• Vectors can be visually modeled using arrows

  • The length of the arrow represents the magnitude of the vector

  • The direction of the arrow represents the direction of the vector

  • The length of the arrows must be proportional their magnitudes

Examples of scalar and vector quantities

• The table below lists some examples of scalar and vector quantities

• Corresponding scalar and vector quantities are aligned where applicable

Table of scalar and vector quantities

Distance and displacement

• Distance is a measure of how far an object travels

• Distance is a scalar quantity with a magnitude but not a direction

• For example, an athlete runs a 300 m race on a 400 m track; the distance traveled by the athlete is 300 m

Distance traveled on a race track

• Displacement is a measure of the change in an object's position

• Displacement is a vector quantity with both magnitude and direction

• For example, the athlete's displacement is 100 m to the right

Distance and displacement on a race track

• Another example is a person hiking in the woods who marks out their route on a map

• The distance traveled is the path they walked

• Their displacement is a straight line arrow drawn from their starting position to their finishing position (so this includes their direction)

Distance and displacement of a hiker

Speed and velocity

• Speed is the distance traveled per unit time

• Speed is a scalar quantity with a magnitude only

• Velocity is the rate of change of displacement

• Velocity is a vector quantity with both magnitude and direction

  • In other words, velocity is speed in a given direction

Mass and weight

• Mass is a measure of the amount of matter in an object

• Mass is a scalar quantity with a magnitude only

• Weight is the gravitational force exerted on an object with mass when placed in a gravitational field

• Force is a vector quantity with both magnitude and direction

Vector notation

• Vectors are given a specific notion with an arrow above the symbol for the given quantity

• Where:

  • = velocity, measured in

  • = initial velocity, measured in

  • = acceleration, measured in

  • = time, measured in

• The magnitude of a vector quantity is represented by parallel lines at either side of the symbol

• Where:

  • = the magnitude of the velocity, measured in

  • = the magnitude of the initial velocity, measured in

  • = the magnitude of the acceleration, measured in

  • = time, measured in

When not to use vector notation

• For vectors in one dimension, the positive or negative value represents the direction

• When vectors are presented along an axis, the axis provides the direction

• Likewise, when the axes are presented in algebraic form, vector notation is not required

• Where:

  • = velocity in the direction, measured in

  • = initial velocity in the direction, measured in

  • = acceleration in the direction, measured in

  • = time, measured in

Derived equation

• In one dimension, the sign of a component completely describes the direction of an object

• For example, in the following equation, the direction is either positive or negative along the -axis

Step 1: Identify the fundamental principle

• Vector components are indicated with arrows above the vector quantities, such as in the following equation:

Step 2: Apply the specific conditions

• In the direction, the vector components become:

  • 

  • 

  • 

• Therefore, the equation in terms of the motion along the axis becomes: