Step by step
The question
Question: find the vector from (0, 0) to (3, 4), and its length.
1 Draw the arrow
Draw the grid.
The arrow starts at (0, 0)
and ends at (3, 4).
Draw the arrow from the start to the end.
We call it v.
2 Count across and up
Across: from x = 0 to x = 3 is right 3.
Up: from y = 0 to y = 4 is up 4.
A vector is written with its two parts: across, then up.
Each part is end minus start.
Careful!
Not start minus end.
That gives ⟨−3, −4⟩: an arrow pointing the other way.
3 minus 0 is 3.
4 minus 0 is 4.
So v = ⟨3, 4⟩.
3 Find the length
Across and up make a right angle.
The arrow is the longest side.
The length ‖v‖: the long side of the right triangle.
Pythagoras: square the two parts, add them, then take the square root.
3² is 3 times 3: 9.
4² is 16.
Write the squares: 9 + 16.
9 plus 16 is 25.
5 times 5 is 25.
So ‖v‖ = 5.
4 Check
Let's check: start at (0, 0) and add the parts.
We land on (3, 4), the end!
And the length.
Square the length: we get 25.
The same as 9 plus 16.
Right!
Let's look back.
First: draw the arrow.
Then: count across and up.
Then: find the length.
Last: check.