# Math help from the Learning Centre

This guide provides useful resources for a wide variety of math topics. It is targeted at students enrolled in a math course or any other Centennial course that requires math knowledge and skills.

## What Is A Vector?

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 "$$\vec{v}$$".

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 $$v_y$$. Then we would write $$\vec{v} = (v_x, v_y)$$. This is the rectangular co-ordinate form of the vector.

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.

Example: Plot the vector $$\vec{v} = (-1,2)$$.

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:

## Alternate Directions

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.

## Practice

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)

1)

2)

3)

4)  $$\vec{v} = (2,4)$$

5)  $$\vec{v} = (-3,-2)$$

## Solving for Components

Instead of being given the components of a vector, sometimes we are given the angle it makes with the x-axis, like in this image.

Once given an angle, as long as we are given either the full length of the vector, the x component, or the y component, we can use trig ratios to solve for the other parts of the vector.

Example 1

Find the x and y components of the following vector:

Solution

Imagine drawing a line from the tip of the vector down to the x-axis. This gives us a right triangle, so we can apply trig ratios to it.

From the definition of the trig ratios, $$\sin 65° = \frac{ \vec{v}_y}{18}$$, so $$18 \sin 65° = \vec{v}_y$$. Putting this into the calculator, we have $$\vec{v}_y \approx 16.3$$.

Similarly, from the definition of cosine, $$\cos 65° = \frac{\vec{v}_x}{18}$$. So we have $$\vec{v}_x = 18 \cos 65° \approx 7.6$$.

This solves the x and y components, so we can write $$\vec{v} = (7.6, 16.3)$$

Example 2

Find the x, y components of the vector with magnitude 60 at angle 225°.

Solution 1

Calculate as before:

$\vec{v}_x = 60 \cos 225° = -42.43$

$\vec{v}_y = 60 \sin 225° = -42.43$

Solution 2

Use the reference angle instead of the full angle.

225° is in the third quadrant, so the reference angle is 225° - 180° = 45°.

Now we can use the Pythagorean theorem (or special triangles) to solve for the components.

$\vec{v}_x = 60 \cos 45° = 42.43$

$\vec{v}_y = 60 \sin 45° = 42.43$

Now, based on the vectors position, put the correct sign on the components:

$\vec{v}_x = -42.43, \vec{v}_y = -42.43$

Note: not using the reference angle is a faster and safer way to get the solution! However, you might choose to use the reference angle because it is more intuitive to see the triangle.