# Components of Vectors

Lesson

What do we know about a vector so far?

Each vector has

• initial point
• terminal point
• magnitude (length)
• direction
• links to right-angled triangles

Knowing some of the above conditions will allow us to calculate the others, and because a vector also has links to trigonometry we can also use the trigonometric ratios to help us with the calculations.

#### Worked Examples

##### Question 1

Given that the vector a, projects from initial point $\left(1,3\right)$(1,3), at an angle of $45^\circ$45°  find the terminal point if the magnitude is $3.5$3.5 units.

The information given to us here, results in the following right-angled triangle image.

The terminal point will have the coordinates $$Using trigonometry we can see \cos45^\circ=\frac{x}{3.5}cos45°=x3.5, so x=3.5\times\cos45^\circx=3.5×cos45° and similarly \sin45^\circ=\frac{y}{3.5}sin45°=y3.5, so y=3.5\times\sin45^\circy=3.5×sin45°. Evaluating these to 22 decimal places we get x=2.47x=2.47 and y=2.47y=2.47. The fact that both x and y are equal make sense because an angle of 45^\circ45° creates an isosceles triangle. Now we can work out the terminal point,$$

## Visualising the components

This applet will help you to visualise the $x$x component and $y$y component.  Remember that it uses the principles of right-angled trigonometry.

#### Worked examples

##### Question 1

Consider the vector with an initial point $\left(2,5\right)$(2,5) and a terminal point $\left(4,8\right)$(4,8).

1. Find the $x$x-component.

2. Find the $y$y-component.

##### Question 2

Plot the vector with an $x$x-component $5$5 and a $y$y-component $9$9.

Use the origin as the starting point for the vector.

##### Question 3

Let $G$G and $H$H be the points $G$G$\left(11,3\right)$(11,3) and $H$H$\left(12,-2\right)$(12,2).

1. Find the vector $\vec{HG}$HG in component form:

$\vec{HG}$HG$=$=$\left(\editable{},\editable{}\right)$(,)

2. What is the exact length of the vector $\vec{HG}$HG?