Vector from (1, 2) to (4, 6) Step by Step

Step by step

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.

Answer