Vector from (2, 1) to (8, 9) Step by Step

Step by step

The question

Question: find the vector from (2, 1) to (8, 9), and its length.

(2, 1) to (8, 9)

1 Draw the arrow

Draw the grid.

The arrow starts at (2, 1)

and ends at (8, 9).

Draw the arrow from the start to the end.

We call it v.

2 Count across and up

Across: from x = 2 to x = 8 is right 6.

Up: from y = 1 to y = 9 is up 8.

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

Each part is end minus start.

v = ⟨8 − 2, 9 − 1⟩

end − start

Careful!

Not start minus end.

That gives ⟨−6, −8⟩: an arrow pointing the other way.

Careful: not ⟨2 − 8, 1 − 9⟩ ✗

8 minus 2 is 6.

9 minus 1 is 8.

So v = ⟨6, 8⟩.

v = ⟨6, 8⟩

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‖ = √6² + 8²

6² is 6 times 6: 36.

6² = 6 × 6 = 36

8² is 64.

8² = 8 × 8 = 64

Write the squares: 36 + 64.

‖v‖ = √36 + 64

36 plus 64 is 100.

‖v‖ = √100

10 times 10 is 100.

10 × 10 = 100

So ‖v‖ = 10.

‖v‖ = 10

4 Check

Let's check: start at (2, 1) and add the parts.

We land on (8, 9), the end!

(2 + 6, 1 + 8) = (8, 9)

And the length.

Square the length: we get 100.

The same as 36 plus 64.

Right!

10 × 10 = 100 and 36 + 64 = 100

Let's look back.

First: draw the arrow.

Then: count across and up.

Then: find the length.

Last: check.

Answer