The fundamental building blocks of linear algebra are *vectors*. Vectors are defined as quantities having both direction and magnitude, compared to *scalar* quantities that only have magnitude. In order to have direction and magnitude, vector quantities consist of two or more elements of data. The *dimensionality* of a vector is determined by the number of numerical elements in that vector. For example, a vector with four elements would have a dimensionality of four.

Let’s take a look at examples of a scalar versus a vector. A car driving at a speed of 40mph is a scalar quantity. Describing the car driving 40mph to the east would represent a two-dimensional vector quantity since it has a magnitude in both the *x* and *y* directions.

Vectors can be represented as a series of numbers enclosed in parentheses, angle brackets, or square brackets. In this lesson, we will use square brackets for consistency. For example, a three-dimensional vector is written as:

```
$v = \begin{bmatrix}
x \\
y \\
z
\end{bmatrix}$
```

The magnitude (or length) of a vector, *||v||*, can be calculated with the following formula:

`$||v|| = \sqrt{\sum\limits_{i=1}^{n} v_{i}^2}$`

This formulates translates to the sum of each vector component squared, which can be also written out as:

`$||v|| = \sqrt{v_{1}^2 + v_{2}^2 + \dots + v_{n}^2}$`

Let’s look at an example. We are told that a ball is traveling through the air and given the velocities of that ball in its *x*, *y*, and *z* directions in a standard Cartesian coordinate system. The velocity component values are:

*x*= -12*y*= 8*z*= -2

Convert the velocities into a vector, and find the total speed of the ball. (Hint: the speed of the ball is the magnitude of the velocity vector!)

Click the dropdown below to check your answers!

## Solution

### Instructions

The applet to the right allows you to play around with the *x* and *y* components of a vector. By clicking on the arrowhead of the red vector you can drag it around the *xy*-plane. You will notice that the magnitude is automatically calculated for you as well. Note that the magnitude is always positive.

Use this applet and/or mathematical calculation to find the magnitudes of the following vectors:

```
$a = \begin{bmatrix}
3 \\
- 4\\
\end{bmatrix}$
```

```
$b = \begin{bmatrix}
0 \\
26 \\
\end{bmatrix}$
```

```
$c = \begin{bmatrix}
26 \\
0 \\
\end{bmatrix}$
```

```
$d = \begin{bmatrix}
-12 \\
13 \\
\end{bmatrix}$
```

Note: It may be hard to get to exact values with the applet, but you can use approximations to help verify your mathematical calculations.