Step by step
The question
Question: find the magnitude and the direction of v = ⟨−3, 4⟩.
1 Draw the arrow
Draw the grid.
The arrow starts at the origin.
It goes left 3 and up 4.
Across 3 and up 4 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: 9 + 16.
9 plus 16 is 25.
5 times 5 is 25.
So ‖v‖ = 5.
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 53.13.
To the nearest tenth, α ≈ 53.1°.
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 up.
θ is a half turn, 180°, take away α.
So θ = 180° − 53.1°.
Careful!
53.1° alone points right and up.
Our arrow points left and up.
Always look at the picture.
180 minus 53.1 is 126.9.
So θ ≈ 126.9°.
So the magnitude is 5 and the direction is 126.9°.
5 Check
Let's check: tan of 126.9° on the calculator is about −1.333.
Up over across, with the signs: 4 over −3 is about −1.333.
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.