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