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Subsection 2.9.1 Vector and Scalar Quantities

In our daily lives, we measure many things using just a single number, such as the temperature of a room or the mass of an object. However, for some physical quantities, knowing “how much” is not enough; we also need to know “which way”. For example, knowing a hospital is \(5\) km away is not helpful unless you also know the direction to travel. This section explores the key differences between quantities that only have magnitude and those that have both magnitude and direction.

Learner Experience 2.9.3.

Work in groups
Imagine it is a normal school day and students are playing football during games time. Suddenly, one student gets injured and needs to be taken to the nearest hospital immediately. A boda boda rider has agreed to help, but he does not know the way. You will use vectors and scalars to plan a safe route.

(a)

Go into the school compound field, choose a starting point, and mark it as point \(\text{A}\) (representing the school gate).

(b)

Standing at point \(\text{A}\text{,}\) locate the north, south, west, and east directions.

(c)

From point \(\text{A}\text{,}\) walk \(20\) steps to the north and mark it as point \(\text{B}\text{.}\) Notice: "20 steps" is a scalar (magnitude only), but "20 steps north" is a vector (magnitude and direction).

(d)

From point \(\text{B}\text{,}\) walk \(15\) steps to the east and mark it as point \(\text{C}\text{.}\)

(e)

On a piece of graph paper, draw a sketch showing your path from point \(\text{A}\) to \(\text{B}\) and then from \(\text{B}\) to \(\text{C}\text{.}\) Use arrows to indicate direction.

(f)

Using your path as reference, draw a clear and simple map to guide the boda boda rider from the school gate (A) to the nearest hospital. Include all turns (left, right, straight) and at least three landmarks (market, police station, church, large tree). Use arrows to show direction and label roads if you know their names.

(g)

Finally, add a compass showing North, South, East, and West to make your directions clearer. Review your map with the group to ensure it is easy to understand and follow.

(h)

Think about why directions (north, east) are necessary in addition to distances (20 steps, 15 steps). Discuss with your group: What is the difference between "walk 20 steps" and "walk 20 steps north"? Which one is a vector and which is a scalar?

Definition 2.9.4.

A vector is a quantity that has both magnitude and direction, whereas a scalar is a quantity which has only magnitude.
Examples of vector quantities include force, velocity, and displacement, while scalar quantities include mass,temperature, and speed.

Key Takeaway 2.9.5.

Table 2.9.6. Difference Between Vector and Scalar Quantities
Feature Vector Quantity Scalar Quantity
Definition Has both magnitude and direction Has only magnitude
Examples Force,Acceleration,Displacement Distance,Temperature,Mass

Example 2.9.7.

Consider the following situations at a school sports day:
  1. A sprinter runs 100 meters in the 100 m race
  2. A discus thrower hurls the discus 60 meters north
  3. A participant’s body temperature is 37°C
  4. A high jumper exerts a force upward to jump
Which of these are vectors and which are scalars?
Solution.
Let’s analyze each situation:
  1. Scalar quantity - The 100 meters tells us only the distance covered, not the direction of travel. Even though we know it’s a race on a track, distance alone is a scalar.
  2. Vector quantity - The displacement "60 meters north" includes both magnitude (60 meters) and direction (north), making it a vector.
  3. Scalar quantity - Temperature has no direction. It’s just a measure of how hot or cold something is, so it’s purely a scalar.
  4. Vector quantity - Force has both magnitude and direction (upward in this case), so it’s a vector.

Example 2.9.8.

A motorcycle travels at 60 km/h. Is this a vector or scalar? Now consider that the same motorcycle travels at 60 km/h due west. Is this different?
Solution.
  1. "60 km/h" alone is a scalar. This is the speed of the motorcycle - it tells us how fast it’s moving but not which direction.
  2. "60 km/h due west" is a vector. This is the velocity of the motorcycle - it has both magnitude (60 km/h) and direction (west). We can write this as \(\overrightarrow{v} = 60 \text{ km/h (west)}\text{.}\)
  3. Key difference: Speed is a scalar quantity (magnitude only), while velocity is a vector quantity (magnitude and direction). This distinction is important in physics and mathematics.

Checkpoint 2.9.9. Scalar or Vector Quantity.

Exercises Exercises

1.

For each of the following quantities, classify them as either a vector or a scalar. Explain your reasoning.
  1. The displacement of a car that traveled 50 km east
  2. The speed of a runner who completed a 100 m sprint in 10 seconds
  3. The acceleration of a motorcycle as it speeds up a hill towards the north
  4. The temperature of water in a cooking pot: 75°C
  5. The force applied to push a door open
  6. The mass of a textbook
Answer.
  1. Vector - it specifies both magnitude (50 km) and direction (east)
  2. Scalar - speed only has magnitude; it doesn’t specify direction
  3. Vector - acceleration has both magnitude and direction (towards the north)
  4. Scalar - temperature is only a magnitude with no direction
  5. Vector - force has magnitude and direction
  6. Scalar - mass is only magnitude with no directional component
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