Vector from (1, 2, 3) to (4, 6, 3) Step by Step

Step by step

The question

Question: find the vector from (1, 2, 3) to (4, 6, 3), and its length.

(1, 2, 3) to (4, 6, 3)

1 Draw it

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

The arrow starts at (1, 2, 3)

and ends at (4, 6, 3).

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 − 1, 6 − 2, 3 − 3⟩

end − start

Careful!

Not start minus end.

That gives an arrow pointing the other way.

Careful: not ⟨−3, −4, 0⟩ ✗

4 minus 1 is 3.

6 minus 2 is 4.

3 minus 3 is 0.

So v = ⟨3, 4, 0⟩.

v = ⟨3, 4, 0⟩

3 Find the length

The length ‖v‖ is Pythagoras with three parts.

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

‖v‖ = √3² + 4² + 0²

3 squared is 9.

4 squared is 16.

0 squared is 0.

‖v‖ = √9 + 16 + 0

Add: 9 plus 16 plus 0 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, 3) and add the parts.

We land on (4, 6, 3), the end!

(1 + 3, 2 + 4, 3 + 0) = (4, 6, 3)

And the length squared is 25, the sum of the squares.

Right!

5² = 25

Let's look back.

First: draw it.

Then: work part by part.

Then: find the length.

Last: check.

Answer