Worked examples
The question
Question: find the magnitude and the direction of v = ⟨3, 3⟩.
v = ⟨3, 3⟩
1 Draw the arrow
Draw the grid.
The arrow starts at the origin.
It goes right 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.
‖v‖ = √32 + 32
Write the squares: 9 + 9.
‖v‖ = √9 + 9
9 plus 9 is 18.
‖v‖ = √18
But 18 has a square number in it: 9.
18 is 9 times 2.
‖v‖ = √9 × 2
A root of a times splits into two roots.
‖v‖ = √9 × √2
√9 is 3.
3 × 3 = 9
So ‖v‖ = 3√2.
That is the exact answer, made simple.
‖v‖ = 3√2
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.
tan α = 33
3 divided by 3 is 1.
3 ÷ 3 = 1
The tan⁻¹ button turns it back into the angle.
α = tan−1(1)
The calculator gives exactly 45.
So α = 45°.
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 right and up.
So θ is just α.
right and up: θ = α
So θ = 45°.
θ = 45°
So the magnitude is 3√2 and the direction is 45°.
‖v‖ = 3√2 and θ = 45°
5 Check
Let's check: tan of 45° on the calculator is 1.
Up over across, with the signs: 3/3 is 1.
The same!
tan 45° = 1, 33 = 1
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.
Answer ‖v‖ = 3√2 and θ = 45°
▶ Watch it solved
The question
Question: find the magnitude and the direction of v = ⟨−3, 3⟩.
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.
‖v‖ = √(−3)2 + 32
Write the squares: 9 + 9.
‖v‖ = √9 + 9
9 plus 9 is 18.
‖v‖ = √18
But 18 has a square number in it: 9.
18 is 9 times 2.
‖v‖ = √9 × 2
A root of a times splits into two roots.
‖v‖ = √9 × √2
√9 is 3.
3 × 3 = 9
So ‖v‖ = 3√2.
That is the exact answer, made simple.
‖v‖ = 3√2
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.
tan α = 33
3 divided by 3 is 1.
3 ÷ 3 = 1
The tan⁻¹ button turns it back into the angle.
α = tan−1(1)
The calculator gives exactly 45.
So α = 45°.
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.
left and up: θ = 180° − α
So θ = 180° − 45°.
θ = 180° − 45°
Careful!
45° alone points right and up.
Our arrow points left and up.
Always look at the picture.
Careful: not θ = 45° ✗
180 minus 45 is 135.
So θ = 135°.
θ = 135°
So the magnitude is 3√2 and the direction is 135°.
‖v‖ = 3√2 and θ = 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!
tan 135° = −1, 3−3 = −1
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.
Answer ‖v‖ = 3√2 and θ = 135°
▶ Watch it solved
The question
Question: find the magnitude and the direction of v = ⟨3, 4⟩.
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.
‖v‖ = √32 + 42
Write the squares: 9 + 16.
‖v‖ = √9 + 16
9 plus 16 is 25.
‖v‖ = √25
5 times 5 is 25.
5 × 5 = 25
So ‖v‖ = 5.
‖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.
tan α = 43
The tan⁻¹ button turns it back into the angle.
α = tan−1(43)
The calculator gives about 53.13.
To the nearest tenth, α ≈ 53.1°.
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.
right and up: θ = α
So θ ≈ 53.1°.
θ ≈ 53.1°
Careful!
Up goes on top, across at the bottom.
Swapped, we would get 36.9°, the angle from the y-axis.
Careful: not tan α = 34 ✗
So the magnitude is 5 and the direction is 53.1°.
‖v‖ = 5 and θ ≈ 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!
tan 53.1° ≈ 1.333, 43 ≈ 1.333
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.
Answer ‖v‖ = 5 and θ ≈ 53.1°
▶ Watch it solved
