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

Step by step

The question

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

(−1, −1) to (1, 5)

1 Draw the arrow

Draw the grid.

The arrow starts at (−1, −1)

and ends at (1, 5).

Draw the arrow from the start to the end.

We call it v.

2 Count across and up

Across: from x = −1 to x = 1 is right 2.

Up: from y = −1 to y = 5 is up 6.

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

Each part is end minus start.

v = ⟨1 − (−1), 5 − (−1)⟩

end − start

Careful!

Not start minus end.

That gives ⟨−2, −6⟩: an arrow pointing the other way.

Careful: not ⟨−1 − 1, −1 − 5⟩ ✗

1 minus (−1) is 2.

5 minus (−1) is 6.

So v = ⟨2, 6⟩.

v = ⟨2, 6⟩

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‖ = √2² + 6²

2² is 2 times 2: 4.

2² = 2 × 2 = 4

6² is 36.

6² = 6 × 6 = 36

Write the squares: 4 + 36.

‖v‖ = √4 + 36

4 plus 36 is 40.

‖v‖ = √40

But 40 has a square number in it: 4.

40 is 4 times 10.

‖v‖ = √4 × 10

A root of a times splits into two roots.

‖v‖ = √4 × √10

√4 is 2.

2 × 2 = 4

So ‖v‖ = 2√10.

That is the exact answer, made simple.

‖v‖ = 2√10

4 Check

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

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

(−1 + 2, −1 + 6) = (1, 5)

And the length.

Square the length: we get 40.

The same as 4 plus 36.

Right!

(2√10)² = 4 × 10 = 40 and 4 + 36 = 40

Let's look back.

First: draw the arrow.

Then: count across and up.

Then: find the length.

Last: check.

Answer