Magnitude and direction of <3, 3> Step by Step

Step by step

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