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: Worksheet 1

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: Worksheet 2

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 \(\vec{h} = \begin{pmatrix} 4 \\ 4 \end{pmatrix}\).


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