Vector from (-2, 3) to (2, -1) Step by Step

Step by step

The question

Question: find the vector from (−2, 3) to (2, −1), and its length.

(−2, 3) to (2, −1)

1 Draw the arrow

Draw the grid.

The arrow starts at (−2, 3)

and ends at (2, −1).

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 = 3 to y = −1 is down 4.

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

Each part is end minus start.

v = ⟨2 − (−2), −1 − 3⟩

end − start

Careful!

Not start minus end.

That gives ⟨−4, 4⟩: an arrow pointing the other way.

Careful: not ⟨−2 − 2, 3 − (−1)⟩ ✗

2 minus (−2) is 4.

−1 minus 3 is −4.

So v = ⟨4, −4⟩.

A minus means left or down.

v = ⟨4, −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‖ = √4² + (−4)²

4² is 4 times 4: 16.

4² = 4 × 4 = 16

(−4)² is 16.

Minus times minus is plus.

(−4)² = (−4) × (−4) = 16

Write the squares: 16 + 16.

‖v‖ = √16 + 16

16 plus 16 is 32.

‖v‖ = √32

But 32 has a square number in it: 16.

32 is 16 times 2.

‖v‖ = √16 × 2

A root of a times splits into two roots.

‖v‖ = √16 × √2

√16 is 4.

4 × 4 = 16

So ‖v‖ = 4√2.

That is the exact answer, made simple.

‖v‖ = 4√2

4 Check

Let's check: start at (−2, 3) and add the parts.

We land on (2, −1), the end!

(−2 + 4, 3 + (−4)) = (2, −1)

And the length.

Square the length: we get 32.

The same as 16 plus 16.

Right!

(4√2)² = 16 × 2 = 32 and 16 + 16 = 32

Let's look back.

First: draw the arrow.

Then: count across and up.

Then: find the length.

Last: check.

Answer