Worked examples
The question
Question: find the vector from (1, 2) to (4, 6), and its length.
(1, 2) to (4, 6)
1 Draw the arrow
Draw the grid.
The arrow starts at (1, 2)
and ends at (4, 6).
Draw the arrow from the start to the end.
We call it v.
2 Count across and up
Across: from x = 1 to x = 4 is right 3.
Up: from y = 2 to y = 6 is up 4.
A vector is written with its two parts: across, then up.
Each part is end minus start.
v = ⟨4 − 1, 6 − 2⟩
end − start
Careful!
Not start minus end.
That gives ⟨−3, −4⟩: an arrow pointing the other way.
Careful: not ⟨1 − 4, 2 − 6⟩ ✗
4 minus 1 is 3.
6 minus 2 is 4.
So v = ⟨3, 4⟩.
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.
‖v‖ = √32 + 42
3² is 3 times 3: 9.
32 = 3 × 3 = 9
4² is 16.
42 = 4 × 4 = 16
Write the squares: 9 + 16.
‖v‖ = √9 + 16
9 plus 16 is 25.
‖v‖ = √25
5 times 5 is 25.
5 × 5 = 25
So ‖v‖ = 5.
‖v‖ = 5
4 Check
Let's check: start at (1, 2) and add the parts.
We land on (4, 6), the end!
(1 + 3, 2 + 4) = (4, 6)
And the length.
Square the length: we get 25.
The same as 9 plus 16.
Right!
5 × 5 = 25 and 9 + 16 = 25
Let's look back.
First: draw the arrow.
Then: count across and up.
Then: find the length.
Last: check.
▶ Watch it solved
The question
Question: find the vector from (3, 5) to (0, 1), and its length.
(3, 5) to (0, 1)
1 Draw the arrow
Draw the grid.
The arrow starts at (3, 5)
and ends at (0, 1).
Draw the arrow from the start to the end.
We call it v.
2 Count across and up
Across: from x = 3 to x = 0 is left 3.
Down: from y = 5 to y = 1 is down 4.
A vector is written with its two parts: across, then up.
Each part is end minus start.
v = ⟨0 − 3, 1 − 5⟩
end − start
Careful!
Not start minus end.
That gives ⟨3, 4⟩: an arrow pointing the other way.
Careful: not ⟨3 − 0, 5 − 1⟩ ✗
0 minus 3 is −3.
1 minus 5 is −4.
So v = ⟨−3, −4⟩.
A minus means left or down.
v = ⟨−3, −4⟩
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.
‖v‖ = √(−3)2 + (−4)2
(−3)² is (−3) times (−3): 9.
Minus times minus is plus.
(−3)2 = (−3) × (−3) = 9
(−4)² is 16.
(−4)2 = (−4) × (−4) = 16
Write the squares: 9 + 16.
‖v‖ = √9 + 16
9 plus 16 is 25.
‖v‖ = √25
5 times 5 is 25.
5 × 5 = 25
So ‖v‖ = 5.
‖v‖ = 5
4 Check
Let's check: start at (3, 5) and add the parts.
We land on (0, 1), the end!
(3 + (−3), 5 + (−4)) = (0, 1)
And the length.
Square the length: we get 25.
The same as 9 plus 16.
Right!
5 × 5 = 25 and 9 + 16 = 25
Let's look back.
First: draw the arrow.
Then: count across and up.
Then: find the length.
Last: check.
▶ Watch it solved
The question
Question: find the vector from (1, 1) to (4, 4), and its length.
(1, 1) to (4, 4)
1 Draw the arrow
Draw the grid.
The arrow starts at (1, 1)
and ends at (4, 4).
Draw the arrow from the start to the end.
We call it v.
2 Count across and up
Across: from x = 1 to x = 4 is right 3.
Up: from y = 1 to y = 4 is up 3.
A vector is written with its two parts: across, then up.
Each part is end minus start.
v = ⟨4 − 1, 4 − 1⟩
end − start
Careful!
Not start minus end.
That gives ⟨−3, −3⟩: an arrow pointing the other way.
Careful: not ⟨1 − 4, 1 − 4⟩ ✗
4 minus 1 is 3.
4 minus 1 is 3.
So v = ⟨3, 3⟩.
v = ⟨3, 3⟩
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.
‖v‖ = √32 + 32
3² is 3 times 3: 9.
32 = 3 × 3 = 9
Write the squares: 9 + 9.
‖v‖ = √9 + 9
9 plus 9 is 18.
‖v‖ = √18
But 18 has a square number in it: 9.
18 is 9 times 2.
‖v‖ = √9 × 2
A root of a times splits into two roots.
‖v‖ = √9 × √2
√9 is 3.
3 × 3 = 9
So ‖v‖ = 3√2.
That is the exact answer, made simple.
‖v‖ = 3√2
4 Check
Let's check: start at (1, 1) and add the parts.
We land on (4, 4), the end!
(1 + 3, 1 + 3) = (4, 4)
And the length.
Square the length: we get 18.
The same as 9 plus 9.
Right!
(3√2)² = 9 × 2 = 18 and 9 + 9 = 18
Let's look back.
First: draw the arrow.
Then: count across and up.
Then: find the length.
Last: check.
▶ Watch it solved
