13.03 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 example

example 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. 

 

Practice questions

Question 1

Question 2

Question 3