Step by step
The question
Question: find the magnitude and the direction of v = ⟨2, −2⟩.
1 Draw the arrow
Draw the grid.
The arrow starts at the origin.
It goes right 2 and down 2.
Across 2 and down 2 make a right triangle.
2 Find the length
The length ‖v‖: the long side of the right triangle.
Pythagoras: square the two parts, add them, then take the square root.
Write the squares: 4 + 4.
4 plus 4 is 8.
But 8 has a square number in it: 4.
8 is 4 times 2.
A root of a times splits into two roots.
√4 is 2.
So ‖v‖ = 2√2.
That is the exact answer, made simple.
3 Find the angle in the triangle
In the right triangle, tan is opposite over adjacent: up over across.
We use the sizes, with no signs.
We call this small angle α.
2 divided by 2 is 1.
The tan⁻¹ button turns it back into the angle.
The calculator gives exactly 45.
So α = 45°.
Here is α on the grid.
4 Which way it points
The direction θ always turns from the right side of the x-axis, going up and round.
The arrow points right and down.
θ is a whole turn, 360°, take away α.
So θ = 360° − 45°.
Careful!
45° alone points right and up.
Our arrow points right and down.
Always look at the picture.
360 minus 45 is 315.
So θ = 315°.
So the magnitude is 2√2 and the direction is 315°.
5 Check
Let's check: tan of 315° on the calculator is −1.
Up over across, with the signs: −2/2 is −1.
The same!
Let's look back.
First: draw the arrow.
Then: find the length.
Then: find the angle in the triangle.
Then: which way it points.
Last: check.