Vector from (3, 2, 1) to (5, 5, 7) Step by Step

Step by step

The question

Question: find the vector from (3, 2, 1) to (5, 5, 7), and its length.

(3, 2, 1) to (5, 5, 7)

1 Draw it

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

The arrow starts at (3, 2, 1)

and ends at (5, 5, 7).

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 = ⟨5 − 3, 5 − 2, 7 − 1⟩

end − start

Careful!

Not start minus end.

That gives an arrow pointing the other way.

Careful: not ⟨−2, −3, −6⟩ ✗

5 minus 3 is 2.

5 minus 2 is 3.

7 minus 1 is 6.

So v = ⟨2, 3, 6⟩.

v = ⟨2, 3, 6⟩

3 Find the length

The length ‖v‖ is Pythagoras with three parts.

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

‖v‖ = √2² + 3² + 6²

2 squared is 4.

3 squared is 9.

6 squared is 36.

‖v‖ = √4 + 9 + 36

Add: 4 plus 9 plus 36 is 49.

‖v‖ = √49

7 times 7 is 49.

7 × 7 = 49

So ‖v‖ = 7.

‖v‖ = 7

4 Check

Let's check: start at (3, 2, 1) and add the parts.

We land on (5, 5, 7), the end!

(3 + 2, 2 + 3, 1 + 6) = (5, 5, 7)

And the length squared is 49, the sum of the squares.

Right!

7² = 49

Let's look back.

First: draw it.

Then: work part by part.

Then: find the length.

Last: check.

Answer