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

Step by step

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.

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.

Answer