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.

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?
graph of three vectors

Optional Problems