Vector from (4, -6) to (-1, 6) Step by Step

Step by step

The question

Question: find the vector from (4, −6) to (−1, 6), and its length.

(4, −6) to (−1, 6)

1 Draw the arrow

Draw the grid.

The arrow starts at (4, −6)

and ends at (−1, 6).

Draw the arrow from the start to the end.

We call it v.

2 Count across and up

Across: from x = 4 to x = −1 is left 5.

Up: from y = −6 to y = 6 is up 12.

A vector is written with its two parts: across, then up.

Each part is end minus start.

v = ⟨−1 − 4, 6 − (−6)⟩

end − start

Careful!

Not start minus end.

That gives ⟨5, −12⟩: an arrow pointing the other way.

Careful: not ⟨4 − (−1), −6 − 6⟩ ✗

−1 minus 4 is −5.

6 minus (−6) is 12.

So v = ⟨−5, 12⟩.

A minus means left or down.

v = ⟨−5, 12⟩

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‖ = √(−5)² + 12²

(−5)² is (−5) times (−5): 25.

Minus times minus is plus.

(−5)² = (−5) × (−5) = 25

12² is 144.

12² = 12 × 12 = 144

Write the squares: 25 + 144.

‖v‖ = √25 + 144

25 plus 144 is 169.

‖v‖ = √169

13 times 13 is 169.

13 × 13 = 169

So ‖v‖ = 13.

‖v‖ = 13

4 Check

Let's check: start at (4, −6) and add the parts.

We land on (−1, 6), the end!

(4 + (−5), −6 + 12) = (−1, 6)

And the length.

Square the length: we get 169.

The same as 25 plus 144.

Right!

13 × 13 = 169 and 25 + 144 = 169

Let's look back.

First: draw the arrow.

Then: count across and up.

Then: find the length.

Last: check.

Answer