Vector from (3, 5) to (0, 1) Step by Step

Step by step

The question

Question: find the vector from (3, 5) to (0, 1), and its length.

(3, 5) to (0, 1)

1 Draw the arrow

Draw the grid.

The arrow starts at (3, 5)

and ends at (0, 1).

Draw the arrow from the start to the end.

We call it v.

2 Count across and up

Across: from x = 3 to x = 0 is left 3.

Down: from y = 5 to y = 1 is down 4.

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

Each part is end minus start.

v = ⟨0 − 3, 1 − 5⟩

end − start

Careful!

Not start minus end.

That gives ⟨3, 4⟩: an arrow pointing the other way.

Careful: not ⟨3 − 0, 5 − 1⟩ ✗

0 minus 3 is −3.

1 minus 5 is −4.

So v = ⟨−3, −4⟩.

A minus means left or down.

v = ⟨−3, −4⟩

3 Find the length

Across and down 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‖ = √(−3)² + (−4)²

(−3)² is (−3) times (−3): 9.

Minus times minus is plus.

(−3)² = (−3) × (−3) = 9

(−4)² is 16.

(−4)² = (−4) × (−4) = 16

Write the squares: 9 + 16.

‖v‖ = √9 + 16

9 plus 16 is 25.

‖v‖ = √25

5 times 5 is 25.

5 × 5 = 25

So ‖v‖ = 5.

‖v‖ = 5

4 Check

Let's check: start at (3, 5) and add the parts.

We land on (0, 1), the end!

(3 + (−3), 5 + (−4)) = (0, 1)

And the length.

Square the length: we get 25.

The same as 9 plus 16.

Right!

5 × 5 = 25 and 9 + 16 = 25

Let's look back.

First: draw the arrow.

Then: count across and up.

Then: find the length.

Last: check.

Answer