# Scalars and Vectors

## Scalars and Vectors Revision

**Scalars and Vectors**

Quantities can be broken down into two categories, **scalars** and** vectors**. In this section we look at the difference between scalars and vectors and some examples of each. We also look at methods of **resolving** and **combining** vectors.

**Difference between Scalars and Vectors**

A** scalar** is a quantity which only has a **magnitude**. The magnitude is the size of the measurement.

Some examples of **scalar** quantities include (but are not limited to):

**Temperature****Charge****Distance****Speed****Time****Energy****Volume****Density****Length**

A **vector** quantity has a **magnitude** and **direction**. Both must be stated for vector quantities.

Some **vector** quantities include:

**Displacement****Velocity****Force****Momentum****Acceleration**

Some of the quantities above are often used incorrectly. For example,** distanc****e** and **displacement **are often used interchangeably when they are two different quantities.

**Distance**is the actual distance an object has moved along the route it has travelled.**Displacement**is the distance an object has travelled directly from start to finish in a straight line. This should include direction of magnitude.

**Addition of Vectors**

The best way to represent a **vector** is by using a scaled arrow. The **length** of the arrow represents the **magnitude** whilst the **directio****n** of the arrow represents the direction.

The best way to show multiple **vectors** is by using a diagram of all the **vectors** in one. The** vector** arrows should be placed top to tail as seen in the example below. The resultant **vector** is found using a straight line from the tail of the first vector, to the head of the final vector. If the two points are in the same place, then the **resultant vecto****r** has zero **magnitude**.

**Example:** A jogger runs 500\: \text{m} east, then 200 \: \text{m} south. What is their displacement?

**[2 marks]**

As this example uses two **vectors **at right angles to each other, we can use calculations to work out the vector:

As the two **vectors** are acting at right angles to each other, Pythagoras’ theorem can be used to calculate the length of the arrow whilst trigonometry can be used to calculate either angle in the triangle.

A^2 + B^2 = C^2

200^2 + 500^2 = C^2

\boldsymbol{C = 538.5 \:} \textbf{m}

If the **vectors** are not acting at right angles to each other, then a **scale diagram** may be used. It is important to note that if this method is used, scales, lengths of arrows and measured angles need to be measured and drawn carefully. The final magnitude of the vector can be measured using a ruler and the direction measured with a protractor.

**Resolving Vectors**

In the opposite process from above, a** vector** can be resolved (separated) into two components; the horizontal and vertical components.

In the **vector** above, we could split the **vector** into its horizontal and vertical components. We can see that if we were to put the vertical and horizontal arrows top to tail, the blue vector is formed.

The horizontal and vertical components of this **vector** can be calculated using the **trigonometr**y above, where F is the **vector** and theta (θ) the angle.

**Example: **Resolve the vector on the right into it’s horizontal and vertical components

**[2 marks]**

Using trigonometry:

- \text{Horizontal component} = 350 \times \cos\left(35\right) = \boldsymbol{290 \:} \textbf{N}
- \text{Vertical component} = 350 \times \sin\left(35\right) = \boldsymbol{200 \:} \textbf{N}

## Scalars and Vectors Example Questions

**Question 1: **Describe the difference between scalar and vector quantities. Your answer should also include examples of each.

**[2 marks]**

A scalar quantity is a quantity that **only has magnitude** while a vector must** also have direction**. Scalars = distance, speed, mass etc. Vectors = displacement, velocity, weight etc….

**Question 2:** A 575 \: \text{N} force acts at 270° to the horizontal. Calculate the horizontal component.

**[2 marks]**

**Question 3:** In the question above, calculate the vertical component of the force.

**[2 marks]**

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