Physics uses measurements to describe the world, but not all measurements carry the same kind of information. A scalar tells how much of something there is, such as mass, time, temperature, or speed. A vector tells both how much and which way, such as displacement, velocity, acceleration, or force.
Knowing the difference matters because direction can completely change the result of a physical situation.
Key Facts
- A scalar has magnitude only, such as 5 kg, 20 s, or 18 m/s.
- A vector has magnitude and direction, such as 12 N east or 4 m/s upward.
- Vector magnitude is written as |A| or A and represents the size of the vector.
- In component form, a two-dimensional vector can be written as A = Ax i + Ay j.
- For perpendicular components, |A| = sqrt(Ax^2 + Ay^2).
- Vectors add by components: Rx = Ax + Bx and Ry = Ay + By.
Vocabulary
- Scalar
- A scalar is a quantity described completely by magnitude with no direction.
- Vector
- A vector is a quantity that has both magnitude and direction.
- Magnitude
- Magnitude is the size or amount of a quantity, such as the length of a vector arrow.
- Component
- A component is the part of a vector that points along a chosen axis, such as the x-axis or y-axis.
- Resultant
- The resultant is the single vector that has the same effect as two or more vectors combined.
Common Mistakes to Avoid
- Treating speed and velocity as the same quantity is wrong because speed is scalar while velocity includes direction.
- Adding vector magnitudes without considering direction is wrong because opposite or angled vectors can partly or completely cancel.
- Forgetting units on vector components is wrong because each component still represents a physical measurement such as meters or newtons.
- Writing a negative vector magnitude is wrong because magnitude is always nonnegative, while the sign belongs to a chosen direction or component.
Practice Questions
- 1 A student walks 6 m east and then 8 m north. Find the magnitude of the displacement vector.
- 2 Two forces act on a box: 15 N to the right and 9 N to the left. Find the net force, including direction.
- 3 A car travels around a circular track and returns to its starting point. Explain why its distance traveled is not zero but its displacement is zero.