Resolving a vector

Unit vectors

Narrated video. Transcript:

Sometimes we only want to say which way, not how much. For that we use a unit vector.

A unit vector has a magnitude of exactly 1, and it has no units. Its only job is to point in a direction.

Along the x, y and z axes we use three special unit vectors: î, ĵ and k̂, read “i hat”, “j hat” and “k hat”. Watch them grow out of the origin, each one unit long, each at right angles to the other two.

Any vector can be written as its magnitude times a unit vector along it. If A has magnitude A, then A = A n̂, where n̂ points along A.

Turn that around and you get the unit vector of any vector: n̂ = A / |A|. Divide a vector by its own length and only its direction is left.

So a displacement of 10 m along the x axis is simply 10 times î, written 10 î m.

Breaking a vector into parts

Narrated video. Transcript:

A batter hits a cricket ball at 20 m/s, at 30° above the ground. How fast is it moving forward, and how fast upward?

Draw the velocity as arrow A from the origin, at angle θ to the x axis. From its tip, drop a dashed line straight down to the x axis.

This makes a right-angled triangle. The side along x is Ax, the x component. The upright side is Ay, the y component. Splitting a vector like this is called resolving it.

From the triangle, Ax = A cos θ and Ay = A sin θ. Put the two parts head to tail and you get A back: A = Ax î + Ay ĵ.

For the cricket ball: Ax = 20 cos 30° ≈ 17.3 m/s forward, and Ay = 20 sin 30° = 10 m/s upward.

It works backwards too. If you know the components, the magnitude is A = √(Ax² + Ay²) and the direction is given by tan θ = Ay / Ax.

Components can be negative. An arrow pointing up and to the left has a negative x component and a positive y component.

In three dimensions, add a third part: A = Ax î + Ay ĵ + Az k̂, and A = √(Ax² + Ay² + Az²).

Components make adding easy. To find A + B, add the x parts and the y parts separately: Rx = Ax + Bx and Ry = Ay + By.

Try it: resolve a vector

Interactive simulation. Things to try:

• Drag the blue tip. Orange is Ax and green is Ay.
• Use the direction slider to set θ exactly, then check A cos θ and A sin θ.
• Move A into each quarter of the plane and watch the signs of Ax and Ay.

Quick check

A force of 10 N acts at 60° to the x axis. What are its x and y components?

1. Fx = 5 N, Fy = 8.66 N
2. Fx = 8.66 N, Fy = 5 N
3. Fx = 10 N, Fy = 10 N
4. Fx = 5 N, Fy = 5 N

Unit vectors

Which of these is a unit vector?

1. î + ĵ
2. (î + ĵ) / √2
3. (î + ĵ) / 2
4. 2î

Summary

• Components: Ax = A cos θ, Ay = A sin θ
• Vector from its components: A = Ax î + Ay ĵ + Az k̂
• Magnitude and direction: A = √(Ax² + Ay²), tan θ = Ay / Ax
• Unit vector along A: n̂ = A / |A|

• î, ĵ and k̂ are unit vectors along x, y and z. They have magnitude 1, no units, and are at right angles to each other.
• Resolving a vector means splitting it into parts along chosen directions, usually the x and y axes.
• A component can be positive, negative or zero, depending on which way the vector points.
• To add vectors, add their components: Rx = Ax + Bx, Ry = Ay + By.