Matrices > Scalar Definition

To understand what a vector is we need to define what a **scalar **is. A **scalar** is a quantity expressed by a single real number. An example of a scalar is length (which can be measured in inches or feet). A scalar has **magnitude**, or the size of a mathematical object. A **vector** is a quantity that is defined by multiple scalars. In addition to magnitude, a vector also has a **direction**.

Consider the following illustration:

The magnitude is located between the initial point (A) and direction (denoted by the lowercase a). Adding the arrow to the line represents the direction. You are most likely to encounter vectors similar to the above on a graph. You need to know how to determine the magnitude and direction between two given points.

## Finding the Magnitude of a Vector

Let’s take a look at line AB on the following graph:

To find the magnitude of vector AB we need to determine the distance between points A and B. Where we are provided with the coordinates of the points we can use the distance formula).

The formula for distance is:

AB = √(x_{2} – x_{1})^{2} + (y_{2} – y_{1})^{2}

**Example**:

Find the magnitude of vector AB where the initial point A is (2, 2) and the end point B is (6, 4).

Plug the values into our distance formula:

AB = √(6-2)^{ 2} + (4-2)^{ 2} = √4^{ 2} + 2^{ 2 }= √16 + 4 = √20

The magnitude of AB = 4.72

## Determining the Direction of a Vector

To find the direction of a vector we measure the angle that the vector makes with a horizontal line. We use the following formula:

tanΘ = y_{2} – y_{1}/x_{2} – x_{1}

**Example**:

What is the direction of vector AB where the initial point A is (2,3) and the end point B is (5,8)

First, we plug the coordinates into our formula for direction:

tanΘ = 8-3/5-2 = 5/3

To find the direction we use the inverse of tan:

Θ = tan^{-1} (5/3)

Vector AB has a direction of 59 degrees.

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