# Introduction to Vectors: Components & How to Draw a Vector

## (What's a Vector?)

Vectors are quantities that are defined by two things:

• their magnitude, and
• their direction.
A couple of examples of vectors are velocity and force.

In 2 dimensions, a vector is described by two components:

• a horizontal component, $$x$$ component
• a vertical component, $$y$$ component

these components are represented as either:

• a column vector: $$\vec{u} = \begin{pmatrix} x \\ y \end{pmatrix}$$
• a row vector: $$\vec{u} = \begin{pmatrix} x & y \end{pmatrix}$$
Both can be used but In this set of notes we'll usually work with column vectors.

### Tutorial: what vectors are, what are components and how to draw them

In this tutorial we learn what a vector is. We learn how to write vectors as both column and row vectors as well as how to represent vectors graphically; that is how to draw vectors.

## Exercise 1

Using gridded paper, draw each of the following vectors:

1. the vector $$\vec{a} = \begin{pmatrix} 5 \\ 2 \end{pmatrix}$$
2. the vector $$\vec{b} = \begin{pmatrix} -3 \\ 4 \end{pmatrix}$$
3. the vector $$\vec{c} = \begin{pmatrix} 6 \\ -3 \end{pmatrix}$$
4. the vector $$\vec{d} = \begin{pmatrix} 0 \\ 5 \end{pmatrix}$$
1. the vector $$\vec{e} = \begin{pmatrix} -2 \\ -5 \end{pmatrix}$$
2. the vector $$\vec{f} = \begin{pmatrix} -4 \\ 0 \end{pmatrix}$$
3. the vector $$\vec{g} = \begin{pmatrix} 1 \\ 6 \end{pmatrix}$$
4. the vector $$\vec{h} = \begin{pmatrix} 5 \\ 5 \end{pmatrix}$$

Note: you can download a sheet of gridded paper here.

Note: this exercise can be downloaded as a worksheet to practice with:

### Solutions

1. For $$\vec{a} = \begin{pmatrix} 5 \\ 2 \end{pmatrix}$$ we find:

2. For $$\vec{b} = \begin{pmatrix} -3 \\ 4 \end{pmatrix}$$ we find:

3. For $$\vec{c} = \begin{pmatrix} 6 \\ -3 \end{pmatrix}$$ we find:

4. For $$\vec{d} = \begin{pmatrix} 0 \\ 5 \end{pmatrix}$$ we find:

5. For $$\vec{e} = \begin{pmatrix} -2 \\ -5 \end{pmatrix}$$ we find:

6. For $$\vec{f} = \begin{pmatrix} -4 \\ 0 \end{pmatrix}$$ we find:

7. For $$\vec{g} = \begin{pmatrix} 1 \\ 6 \end{pmatrix}$$ we find:

8. For $$\vec{h} = \begin{pmatrix} 5 \\ 5 \end{pmatrix}$$ we find:

### Tutorial: How to find a vector's components

In this tutorial we learn what a vector is. We learn how to write vectors as both column and row vectors as well as how to represent vectors graphically; that is how to draw vectors.

## Exercise 2

Writing your answers as column vectors, find the coordinates of each of the vectors drawn here:

1. vector $$\vec{a}$$:

2. vector $$\vec{b}$$:

3. vector $$\vec{c}$$:

4. vector $$\vec{d}$$:

5. vector $$\vec{e}$$:

6. vector $$\vec{f}$$:

7. vector $$\vec{g}$$:

8. vector $$\vec{h}$$:

Note: this exercise can be downloaded as a worksheet to practice with:

## Solution Without Working

1. We find vector $$\vec{a} = \begin{pmatrix} 2 \\ 5 \end{pmatrix}$$.

2. We find vector $$\vec{b} = \begin{pmatrix} 5 \\ -1 \end{pmatrix}$$.

3. We find vector $$\vec{c} = \begin{pmatrix} 3 \\ 2 \end{pmatrix}$$.

4. We find vector $$\vec{d} = \begin{pmatrix} 0 \\ -3 \end{pmatrix}$$.

5. We find vector $$\vec{e} = \begin{pmatrix} -5 \\ 0 \end{pmatrix}$$.

6. We find vector $$\vec{f} = \begin{pmatrix} -4 \\ 5 \end{pmatrix}$$.

7. We find vector $$\vec{g} = \begin{pmatrix} 4 \\ -6 \end{pmatrix}$$.

8. We find vector $$\vec{h} = \begin{pmatrix} 4 \\ 4 \end{pmatrix}$$.

Scan this QR-Code with your phone/tablet and view this page on your preferred device.

### Subscribe to Our Channel

Subscribe Now and view all of our playlists & tutorials.