Chapter 2: Vectors

2.2 Representation

Visually, vectors are represented as arrows such as in the figure below.

An arrow in quadrant I of an x-y coordinate system. The end of the arrow is at the origin and the arrow points up and to the right. The arrow is labeled with a capital letter A with a small arrow over the letter. The angle between the x axis and the arrow is labeled with the Greek letter theta. A dashed line extends from the tip of the arrow straight down to the x axis; this line is labeled with a capital A with a subscript y. The segment of the x axis between the origin and this dashed line is labeled with a capital A with a subscript x.
Graphical representation of vector A

Symbolically, a vector is notated with an arrow on top of it when hand-written. The symbol \(\vec{A}\) would be read as “vector A.” When typed, we use a bold-faced font. So, the symbol A would also be read as “vector A.”

The length of the arrow gives the magnitude of the vector. Magnitude is a scalar quantity. The magnitude of a vector A can be written as \(|\vec{A}|\), \(|\mathbf{A}|\), \(|A|\), or simply \(A\) (just the letter, neither bold-faced nor with an arrow on top). When written as \(A\), you need to consider the context to recognize if you are working with the magnitude of a particular vector. In this chapter, I will explicitly note a vector’s magnitude using absolute value bars. In future chapters, I will use the simpler and more common notation without absolute value bars or an arrow.

Now imagine that you build an arrow out of planks of wood. You then hold the arrow up at an angle, and shine a flashlight directly down from above the arrow*. You would see a shadow cast along the ground, under the arrow.

Then pretend that you could do that with the arrow in figure above. You would see its shadow cast along the x-axis. The length of this arrow would give the \(x\) component of the vector. The components of a vector tell you how much that vector points in a particular direction. For a vector in lying in a 2-dimensional plane, using \(x\) and \(y\) Cartesian coordinates, we write out the vector as follows:

\[ \vec{A} = A_x\hat{x} + A_y\hat{y} \tag{2.1} \]

This is called being written in component form, and is a very common way you will see vectors written out. The components \(A_x\) and \(A_y\) (pronounced “A ex” and “A why” or “A sub ex” and “A sub why”) are scalars. The symbols \(\hat{x}\) and \(\hat{y}\) (pronounced “ex hat” and “why hat”) represent vectors with a magnitude of 1, often referred to as unit vectors, which point in the direction of the x-axis and y-axis respectively.** The vector \(\vec{A}\) basically gives directions saying “go \(A_x\) units in the \(x\) direction, and \(A_y\) units in the \(y\) direction.

Example 2.3

You may use the interactive below to see many examples of vectors represented as arrows, and with the corresponding vector notation. Use the sliders at the top to adjust the x and y components of the vector.

*You can do this right at your desk! Use your pencil to represent the vector, and shine a flashlight (which your cell phone probably has) straight down from above.

**You may see the unit vectors \(\hat{x}\) and \(\hat{y}\) written as \(\hat{\imath}\) and \(\hat{\jmath}\). There are very interesting reasons for this that relate to the history of our understanding of both vectors and imaginary numbers.