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 right 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 right and up.
So θ is just α.
So θ ≈ 53.1°.
Careful!
Up goes on top, across at the bottom.
Swapped, we would get 36.9°, the angle from the y-axis.
So the magnitude is 5 and the direction is 53.1°.
5 Check
Let's check: tan of 53.1° on the calculator is about 1.333.
Up over across, with the signs: 4/3 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.