Vector from (0, -2) to (5, 0) Step by Step

Step by step

The question

Question: find the vector from (0, −2) to (5, 0), and its length.

(0, −2) to (5, 0)

1 Draw the arrow

Draw the grid.

The arrow starts at (0, −2)

and ends at (5, 0).

Draw the arrow from the start to the end.

We call it v.

2 Count across and up

Across: from x = 0 to x = 5 is right 5.

Up: from y = −2 to y = 0 is up 2.

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

Each part is end minus start.

v = ⟨5 − 0, 0 − (−2)⟩

end − start

Careful!

Not start minus end.

That gives ⟨−5, −2⟩: an arrow pointing the other way.

Careful: not ⟨0 − 5, −2 − 0⟩ ✗

5 minus 0 is 5.

0 minus (−2) is 2.

So v = ⟨5, 2⟩.

v = ⟨5, 2⟩

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‖ = √5² + 2²

5² is 5 times 5: 25.

5² = 5 × 5 = 25

2² is 4.

2² = 2 × 2 = 4

Write the squares: 25 + 4.

‖v‖ = √25 + 4

25 plus 4 is 29.

‖v‖ = √29

29 is not a square number.

So we keep √29.

That is the exact answer.

‖v‖ = √29

4 Check

Let's check: start at (0, −2) and add the parts.

We land on (5, 0), the end!

(0 + 5, −2 + 2) = (5, 0)

And the length.

Square the length: we get 29.

The same as 25 plus 4.

Right!

√29 × √29 = 29 and 25 + 4 = 29

Let's look back.

First: draw the arrow.

Then: count across and up.

Then: find the length.

Last: check.

Answer