Vector from (2, -1, 4) to (4, 1, 5) Step by Step

Step by step

The question

Question: find the vector from (2, −1, 4) to (4, 1, 5), and its length.

(2, −1, 4) to (4, 1, 5)

1 Draw it

Draw the three axes: x toward us, y right, z up.

The arrow starts at (2, −1, 4)

and ends at (4, 1, 5).

Draw the arrow from the start to the end.

We call it v.

2 Work part by part

Each part is end minus start: x minus x, y minus y, z minus z.

v = ⟨4 − 2, 1 − (−1), 5 − 4⟩

end − start

Careful!

Not start minus end.

That gives an arrow pointing the other way.

Careful: not ⟨−2, −2, −1⟩ ✗

4 minus 2 is 2.

1 minus (−1) is 2.

5 minus 4 is 1.

So v = ⟨2, 2, 1⟩.

v = ⟨2, 2, 1⟩

3 Find the length

The length ‖v‖ is Pythagoras with three parts.

Square each part, add, then take the square root.

‖v‖ = √2² + 2² + 1²

2 squared is 4.

1 squared is 1.

‖v‖ = √4 + 4 + 1

Add: 4 plus 4 plus 1 is 9.

‖v‖ = √9

3 times 3 is 9.

3 × 3 = 9

So ‖v‖ = 3.

‖v‖ = 3

4 Check

Let's check: start at (2, −1, 4) and add the parts.

We land on (4, 1, 5), the end!

(2 + 2, −1 + 2, 4 + 1) = (4, 1, 5)

And the length squared is 9, the sum of the squares.

Right!

3² = 9

Let's look back.

First: draw it.

Then: work part by part.

Then: find the length.

Last: check.

Answer