Math help from the Learning Centre

What is a vector? A vector is a combination of two quantities, a magnitude (an amount), and a direction. When written out, they are usually written in bold or with an arrow overtop, so if we have a vector "v", it would be written as "v" or "\( \vec{v} \)". We can also perform operations with vectors like addition, subtraction, multiplication, etc. However, this page only covers the basics.

The most common way to see vector quantities written (in 2 dimensions) is with rectangular coordinates, like the regular grid you see in math plots. In rectangular coordinates, we can write our vector, \( \vec{v} \), in terms of its x and y components which we will label \( v_x \) and \( v_y \). The most common forms you'll see are:

• Point form: \(  \vec{v} = (v_x,v_y) \)  Here, the vector is written as you would any point in rectangular coordinates.
• Matrix form: \(  \vec{v} = \begin{bmatrix} v_x \\ v_y \end{bmatrix} \)  This form is like point form except using matrix notation in case you are doing matrix calculations.
• Unit vector form: \(  \vec{v} = \hat{i} v_x + \hat{j} v_y \)  Here, the vector is written in terms of the x and y unit vectors, \( \hat{i} \) and \( \hat{j} \).

These forms are all useful for different types of calculations, however, for basic introductions, point form is the most familiar and intuitive so it will be used for the rest of this help page.

Visually, you can picture this vector like the following image, where \( v_x \) can point anywhere on the x axis and \( v_y \) can point anywhere on the y axis.

In the following graphs, it's important to note that the vector isn't just the end point, it's the whole line.

Solution:

Here, our \( v_x = -1\) meaning left 1, and our \( v_y = 2\)  meaning up 2, so we would plot \( \vec{v} \) like this:

We can also describe vectors using directions like north, east, south and west. To represent these directions, we can treat the grid of these plots like a compass, so north is the +y direction (up), east is the +x direction (right), south is the -y direction (down), and west is the -x direction (left).

An example of this would be saying "I am traveling 55km/h southward". Here, "-55km/h" would be the magnitude and "southward" would be the direction. To graph a vector given like this, the magnitude represents the length of the arrow and the direction decides where the arrow points. So, if we use the grid like a compass, \( \vec{v} = \) "55km/h southward" would look like this:

It is also worth noting that these vectors are relative to some reference point. With rectangular coordinates, they are relative to the origin point (0,0). On the other hand, the "I am traveling 55km/h southward" example is relative to the north pole because "southward" is a compass direction and compasses use the north pole as a reference point.

As a quick exercise, try drawing the following vectors:

1) \((4,0)\)

2) 14km/h westward

3) \((2,-2)\)

 

Now, try labeling the following vectors in their rectangular coordinate form:

4)

                                                                                                 

 

5)

                                                                                                 

 

 

 

Answers:

1)

 

2)

 

3)

 

4)  \( \vec{v} = (2,4) \)

5)  \( \vec{v} = (-3,-2) \)