\(\def \u#1{\,\mathrm{#1}}\) \(\def \us#1{\,\mathrm{\scriptsize #1}}\) \(\def \abs#1{\left|#1\right|}\) \(\def \ast{*}\) \(\def \deg{^{\circ}}\) \(\def \tau{\uptau}\) \(\def \ten#1{\times 10^{#1}}\) \(\def \redcancel#1{{\color{red}\cancel{#1}}}\) \(\def \BLUE#1{{\color{blue} #1}}\) \(\def \RED#1{{\color{red} #1}}\) \(\def \PURPLE#1{{\color{purple} #1}}\) \(\def \th#1,#2{#1,\!#2}\) \(\def \lshift#1#2{\underset{\Leftarrow\atop{#2}}#1}}\) \(\def \rshift#1#2{\underset{\Rightarrow\atop{#2}}#1}}\) \(\def \dotspot{{\color{lightgray}{\circ}}}\) \(\def \ccw{\circlearrowleft}\) \(\def \cw{\circlearrowright}\)
Appendix A: Vectors and Trigonometry
9.

Component Form

If we are given the magnitude of a vector $\vec v$ and its angle from one of the cardinal directions, then we can draw the vector as the sum of two components, one along the $x$ axis and one along the $y$ axis. The two components and the vector itself form a right triangle with the vector as the hypotenuse, and we can use trigonometry to find the lengths of the components $v_x$ and $v_y$. When the angle is measured from the x-axis, then $v_x$ is the adjacent side and so $\cos\theta = {v_x/v}$ (where $v$ is shorthand for the magnitude $|\vec v|$ while $\sin\theta={v_y/v}$. If the angle is measured from the $y$ axis, however, the roles are flipped, with cosine using $v_y$ and sine using $v_x$. Note that these calculations only give us the lengths of the component vectors, not their directions. For instance, the vector shown in this figure points up and to the left, so this vector would be written $(-v\cos\theta,v\sin\theta)$ or $(-v\sin\phi,v\cos\phi)$.

If the magnitude is \(v\) and the angle is \(\theta\), then the components will always have one of these forms:

$$\begin{align*} (\pm v\cos\theta, \pm v\sin\theta) \hbox{or}\, (\pm v\sin\theta, \pm v \cos \theta)\\ \end{align*}$$

In other words, we need to determine a) which component (x or y) gets the cosine, and which the sine; and b) whether each component is positive or negative. Here's a method for doing it:

  1. Write a set of parentheses with a comma in between, leaving plenty of blank space for writing.
  2. 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".
  3. 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.
  4. If the vector generally points in the +x direction (usually to the right), then the first term gets a +; to the left, it gets a \(-\).
  5. Similarly, if the vector generally points in the +y direction (usually upward), the second term gets a +; downward, \(-\).

Of course, once you've had some practice you'll be able to write this at once from left to right.

And remember

"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:

And here is a little quiz game that lets you practice.

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.