Step by step
The question
Question: find the magnitude and the direction of v = ⟨8, −6⟩.
1 Draw the arrow
Draw the grid.
The arrow starts at the origin.
It goes right 8 and down 6.
Across 8 and down 6 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: 64 + 36.
64 plus 36 is 100.
10 times 10 is 100.
So ‖v‖ = 10.
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.
We can make 6/8 simpler.
It is the same as 3/4.
The tan⁻¹ button turns it back into the angle.
The calculator gives about 36.87.
To the nearest tenth, α ≈ 36.9°.
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° − 36.9°.
Careful!
36.9° alone points right and up.
Our arrow points right and down.
Always look at the picture.
360 minus 36.9 is 323.1.
So θ ≈ 323.1°.
So the magnitude is 10 and the direction is 323.1°.
5 Check
Let's check: tan of 323.1° on the calculator is about −0.75.
Up over across, with the signs: −6/8 is about −0.75.
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.