\(\def \u#1{\,\mathrm{#1}}\) \(\def \abs#1{\left|#1\right|}\) \(\def \ast{*}\) \(\def \deg{^{\circ}}\) \(\def \tau{\uptau}\) \(\def \ten#1{\times 10^{#1}}\) \(\def \redcancel#1{{\color{red}\cancel{#1}}}\) \(\def \BLUE#1{{\color{blue} #1}}\) \(\def \RED#1{{\color{red} #1}}\) \(\def \PURPLE#1{{\color{purple} #1}}\) \(\def \th#1,#2{#1,\!#2}\) \(\def \lshift#1#2{\underset{\Leftarrow\atop{#2}}#1}}\) \(\def \rshift#1#2{\underset{\Rightarrow\atop{#2}}#1}}\) \(\def \dotspot{{\color{lightgray}{\circ}}}\) \(\def \ccw{\circlearrowleft}\) \(\def \cw{\circlearrowright}\)
Appendix B: Vectors
6.

Component Form

If we are given a vector's magnitude and its angle, then we can use Trigonometric Functions to write its components. If the magnitude is \(v\) and the angle is \(\theta\), then the components will always have one of two forms: $$(\pm v\cos\theta, \pm v\sin\theta) \qquad \hbox{or} \qquad (\pm v\sin\theta, \pm v \cos \theta)$$

Here's how you do it, in the conventional basis:

  1. Find the magnitude of the vector (it might be a number or a variable), and write it in both places. Write the angle in both places as well, leaving room for "sin" or "cos".
  2. Figure out which direction the angle is measured from. If it is measured from the horizontal axis, then the first term gets the cosine, and the second gets the sine. If it is measured from the vertical axis, then the second term gets the cosine, and the first gets the sine.
  3. If the vector generally points to the right, then the first term gets a +; to the left, it gets a \(-\).
  4. Similarly, if the vector generally points upward, the second term gets a +; downward, \(-\).

Get used to saying

"The angle is measured from the \(x\), so the \(x\) gets the cosine."

Here are a couple examples.

Example

(no alternate text)
  1. The magnitude is 4 and the angle is 30°, so we fill them both in.
  2. The angle is measured from the \(x\) axis, so the \(x\) term gets the cosine.
  3. The vector is pointing to the right, so the first term is positive.
  4. The vector is pointing upward, so the second term is positive as well.

Thus, \((+4\cos 30\deg,+4\sin 30\deg)\). (I recommend always including the sign in vector tuples, even when they are positive.)

See if you can follow the same reasoning for this vector:

Optional

Mathematicians and physicists often write a vector like (+3,-5) in the form \(3\hat i-5\hat j\) or \(3\hat x-5\hat y\). The symbol \(\hat x\) is called a unit vector, and is simply a vector that points in the \(+x\) direction and has a length of 1. Thus \(3\hat x\) is a vector which also points in the \(+x\) direction but with a length of 3. You won't need to know that in this book, but if you take another course that uses vectors, or you refer to another textbook, you may see this notation.