Step by step
The question
Question: find the magnitude and the direction of v = ⟨−5, −12⟩.
1 Draw the arrow
Draw the grid.
The arrow starts at the origin.
It goes left 5 and down 12.
Across 5 and down 12 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: 25 + 144.
25 plus 144 is 169.
13 times 13 is 169.
So ‖v‖ = 13.
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.
The tan⁻¹ button turns it back into the angle.
The calculator gives about 67.38.
To the nearest tenth, α ≈ 67.4°.
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 left and down.
θ is a half turn, 180°, and then α more.
So θ = 180° + 67.4°.
Careful!
67.4° alone points right and up.
Our arrow points left and down.
Always look at the picture.
180 plus 67.4 is 247.4.
So θ ≈ 247.4°.
So the magnitude is 13 and the direction is 247.4°.
5 Check
Let's check: tan of 247.4° on the calculator is about 2.4.
Up over across, with the signs: −12 over −5 is about 2.4.
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.