What do we know about a vector so far?
Each vector has
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.
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 $$
Using trigonometry we can see $\cos45^\circ=\frac{x}{3.5}$cos45°=x3.5, so $x=3.5\times\cos45^\circ$x=3.5×cos45° and similarly $\sin45^\circ=\frac{y}{3.5}$sin45°=y3.5, so $y=3.5\times\sin45^\circ$y=3.5×sin45°. Evaluating these to $2$2 decimal places we get $x=2.47$x=2.47 and $y=2.47$y=2.47. The fact that both x and y are equal make sense because an angle of $45^\circ$45° creates an isosceles triangle.
Now we can work out the terminal point, $$
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.
Consider the vector with an initial point $\left(2,5\right)$(2,5) and a terminal point $\left(4,8\right)$(4,8).
Find the $x$x-component.
Find the $y$y-component.
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.
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).
Find the vector $\vec{HG}$→HG in component form:
$\vec{HG}$→HG$=$=$\left(\editable{},\editable{}\right)$(,)
What is the exact length of the vector $\vec{HG}$→HG?