## Vectors: Component Form Notation

### Unit Vectors

A unit vector is a vector with the following characteristics:

- dimensionless (no units)
- magnitude of one

Any vector can be written as the product of its magnitude times its unit vector, \(\vec{a} = a\hat{a}\). That means the unit vector for \(\vec{a}\) can be found from:

$$ \hat{a} = \frac{\vec{a}}{a} $$### Vectors: Component Form Notation

Component Form is representing vectors as a linear combination of vectors that are parallel to the x, y, and z axes. These are created by multiplying components to the Cartesian basis vectors.

$$\vec{a} = a_x\hat{i} + a_y\hat{j} + a_z\hat{k} $$ The magnitude of the components \(a_x, a_y, a_z\) gives the magnitude of that specific vector and the sign indicates the direction. The set of components are exactly the numbers in the ordered set notation.### Converting Vectors to and from Component Form

Let's take a look on how to go back and forth between component form and descriptive form. We'll also look at the ordered set notation.

### Vector Arithmetic in Component Form

Doing arithmetic in component form is a lot like doing it in ordered set notation. For adding, add the components. For scalar multiplication, multiply each component by the scalar. The key is to keep the components separate.

### Exercises

*Do Now!* Do these exercises immediately.

*Not Now!* Do these after you start to forget the topic, say in a week.

*More!* More exercises if you want. Maybe review before a test.

### Required Problems

### Solution

1. In the figure below, the magnitudes are given by \(A = 80\), \(B = 100\), and \(C = 50\). What vector, when added to all the vectors shown will give the zero vector?