Step by step
The question
Question: find the magnitude and the direction of v = ⟨−3, 3⟩.
1 Draw the arrow
Draw the grid.
The arrow starts at the origin.
It goes left 3 and up 3.
Across 3 and up 3 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 + 9.
9 plus 9 is 18.
But 18 has a square number in it: 9.
18 is 9 times 2.
A root of a times splits into two roots.
√9 is 3.
So ‖v‖ = 3√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.
3 divided by 3 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 left and up.
θ is a half turn, 180°, take away α.
So θ = 180° − 45°.
Careful!
45° alone points right and up.
Our arrow points left and up.
Always look at the picture.
180 minus 45 is 135.
So θ = 135°.
So the magnitude is 3√2 and the direction is 135°.
5 Check
Let's check: tan of 135° on the calculator is −1.
Up over across, with the signs: 3 over −3 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.