Introduction to Vectors and Scalars

Published: May 16th, 2020

The main difference between vectors and scalars is in their definition. Vectors are measurements of magnitude with direction of movement, and Scalars are measurements of only magnitude WITHOUT direction. 

Note that magnitude and size are interchangeable in this context. For example, when saying a block is moving at 10 km/h [Left], the magnitude is 10 km/h. 

Vectors:

Figure 1: A block moving 10 meters to the left.

Let's take the example of moving a block in figure 1 to the left. This example defines the blocks that are moving 10 meters to the left. This example clearly defines the magnitude (10 m) and direction (left). Therefore, it is clear that this is an example of a vector. This example shows the displacement of the block, defined numerically as 10 meters left. 

If we were to add a time on how long it took the block to reach that final position, we can find velocity. For example, if the block took 5 seconds to reach the final position, we can find velocity by the following equation: velocity = Displacement / time = 10 m [Left]/ 5s ⇒ 2m/s [Left].

Scalars

Looking at figure 1, if the example had not defined the direction of movement (and not explained the initial and final positions), then it would be a scalar. An example of a scalar would be speed limits on the streets. For example, when the signs say “50 km/h” they define the speed and not the direction of movement. 

Relationship between Displacement, velocity and acceleration:

  • Displacement, velocity and acceleration are all vectors as they define specific directions of movement. They have a simple relationship where the rate of change of displacement = velocity, and the rate of change of velocity = acceleration. Therefore, taking the derivative of displacement gives you velocity and the derivative of velocity gives you acceleration. 
  • The main thing to notice about acceleration is that it is the rate of change of velocity. When you’re in a car and are moving at a constant velocity, there is no acceleration. However, when you speed up or slow down, there is a change of velocity, and therefore a change in acceleration. The equation for average acceleration is as follows: 

\(\Huge{\overline{a} = \frac{v-v_o}{t}=\frac{\Delta v}{\Delta t}}\)

Where  \(v\) = final velocity, and  \(v_o\) = initial velocity. 

Knowing this equation, what would change if the change in time got smaller? What if the change in time was 0.005s (imagine driving a car and only pressing break for 0.005 s)? The main thing here is that the acceleration would be instantaneous acceleration. This type of acceleration would look like a tangent on a velocity time graph and will be explored in the next lecture. 

 

In brief:

  Vector Scalar
Definition Magnitude AND Direction (65 km/h right) Magnitude only ( 65km/h )
Example Displacement (10 meters [right]) Distance (10 meters)
  Velocity (displacement / time)  Speed (Distance / time)

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