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Components of Vectors


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 $x=3.5\times\cos45^\circ$x=3.5×cos45° and similarly $\sin45^\circ=\frac{y}{3.5}$sin45°=y3.5, so $y=3.5\times\sin45^\circ$y=3.5×sin45°.  Evaluating these to $2$2 decimal places we get $x=2.47$x=2.47 and $y=2.47$y=2.47.  The fact that both x and y are equal make sense because an angle of $45^\circ$45° 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.

  1. Loading Graph...

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:


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

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