Magnitude and direction of <2, 2> Step by Step

Step by step

The question

Question: find the magnitude and the direction of v = ⟨2, 2⟩.

v = ⟨2, 2⟩

1 Draw the arrow

Draw the grid.

The arrow starts at the origin.

It goes right 2 and up 2.

Across 2 and up 2 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‖ = √2² + 2²

Write the squares: 4 + 4.

‖v‖ = √4 + 4

4 plus 4 is 8.

‖v‖ = √8

But 8 has a square number in it: 4.

8 is 4 times 2.

‖v‖ = √4 × 2

A root of a times splits into two roots.

‖v‖ = √4 × √2

√4 is 2.

2 × 2 = 4

So ‖v‖ = 2√2.

That is the exact answer, made simple.

‖v‖ = 2√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 = 22

2 divided by 2 is 1.

2 ÷ 2 = 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 2√2 and the direction is 45°.

‖v‖ = 2√2 and θ = 45°

5 Check

Let's check: tan of 45° on the calculator is 1.

Up over across, with the signs: 2/2 is 1.

The same!

tan 45° = 1, 22 = 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