# Mechanics

calculus-based physics for scientists and engineers

## Vectors: Component Form Notation

### Unit Vectors

A unit vector is a vector with the following characteristics:

1. dimensionless (no units)
2. magnitude of one

3. 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.

### 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?