Worked examples
The question
Question: find u + v, when u = ⟨3, −2⟩ and v = ⟨−1, 4⟩.
u = ⟨3, −2⟩ and v = ⟨−1, 4⟩
1 Draw tip to tail
Draw the grid.
Draw u from the origin.
Put the start of v at the tip of u.
Tip to tail.
The answer goes from the very start to the very end.
2 Add the parts
Add the across parts, and the up parts.
Each arrow adds its own across and up.
u + v = ⟨3 + (−1), −2 + 4⟩
Adding a negative is the same as taking away.
+ (−) = −
3 plus (−1) is 2.
−2 plus 4 is 2.
So u + v = ⟨2, 2⟩.
u + v = ⟨2, 2⟩
3 Check
Let's check on the grid.
The red arrow ends at (2, 2).
That is right 2 and up 2 from the start: ⟨2, 2⟩.
The same!
tip (2, 2) → ⟨2, 2⟩
Let's look back.
First: draw tip to tail.
Then: add the parts.
Last: check.
▶ Watch it solved
The question
Question: find u − v, when u = ⟨3, −2⟩ and v = ⟨−1, 4⟩.
u = ⟨3, −2⟩ and v = ⟨−1, 4⟩
1 Draw tip to tail
Draw the grid.
Draw u from the origin.
Taking away v is adding −v: the arrow turned around.
Put the start of −v at the tip of u.
Tip to tail.
The answer goes from the very start to the very end.
2 Take away the parts
Take away the across parts, and the up parts.
Taking away an arrow turns it around, so its parts change sign.
u − v = ⟨3 − (−1), −2 − 4⟩
Taking away a negative is the same as adding.
− (−) = +
Careful!
Minus a negative is plus.
3 minus (−1) is 3 plus 1.
Careful: not 3 − (−1) = 2 ✗
3 minus (−1) is 4.
−2 minus 4 is −6.
So u − v = ⟨4, −6⟩.
u − v = ⟨4, −6⟩
3 Check
Let's check on the grid.
The red arrow ends at (4, −6).
That is right 4 and down 6 from the start: ⟨4, −6⟩.
The same!
tip (4, −6) → ⟨4, −6⟩
Let's look back.
First: draw tip to tail.
Then: take away the parts.
Last: check.
▶ Watch it solved
The question
Question: find 2u + v, when u = ⟨2, 1⟩ and v = ⟨1, −3⟩.
u = ⟨2, 1⟩ and v = ⟨1, −3⟩
1 Multiply by 2
2u is 2 times each part of u.
2u = ⟨2 × 2, 2 × 1⟩
So 2u = ⟨4, 2⟩.
2u = ⟨4, 2⟩
2 Draw tip to tail
Draw the grid.
Draw 2u from the origin.
Put the start of v at the tip of 2u.
Tip to tail.
The answer goes from the very start to the very end.
3 Add the parts
Add the across parts, and the up parts.
Each arrow adds its own across and up.
2u + v = ⟨4 + 1, 2 + (−3)⟩
Adding a negative is the same as taking away.
+ (−) = −
4 plus 1 is 5.
2 plus (−3) is −1.
So 2u + v = ⟨5, −1⟩.
2u + v = ⟨5, −1⟩
4 Check
Let's check on the grid.
The red arrow ends at (5, −1).
That is right 5 and down 1 from the start: ⟨5, −1⟩.
The same!
tip (5, −1) → ⟨5, −1⟩
Let's look back.
First: multiply by 2.
Then: draw tip to tail.
Then: add the parts.
Last: check.
▶ Watch it solved
