Episode title: Craving Components
Preview
Now we know how to give DOT directions to their snacks in two different ways using vectors. First, we learned how to give them directions in rectangular components $\langle V_x, V_y \rangle$ where we told them how far to walk in the $x$-direction and in the $y$-direction. Then we learned how to translate from rectangular form to polar form, where we told DOT which direction to walk using the angle $\theta_V$ and how far to walk using the length $|\vec{V}|$.
DOT needs to log their journeys in rectangular components, even if we give them directions in polar form. In this episode, we learn how to translate from polar form back to rectangular form.
Play
[DOT game still in development!]
Practice
Sometimes it’s easy to guess $V_x$ and $V_y$ when you know the magnitude $|\vec{V}|$ and the standard angle $\theta_V$ of a vector $\vec{V}$, but usually we’ll need to perform a calculation. Since we used trigonometry (in the form of the inverse tangent function $\tan^{-1}$) when we were translating from rectangular coordinates to polar coordinates, it might not surprise you that we’ll use trigonometry again to translate from polar coordinates to rectangular coordinates:
$V_x = |\vec{V}|\cos(\theta_V)$
$V_y = |\vec{V}|\sin(\theta_V)$.
Again, make sure your calculator is set to degree mode before you take sine or cosine of your vector’s angle.
If you have taken trigonometry before, you have probably seen the formulas $\sin(\theta) = \dfrac{O}{H}$ and $\cos(\theta) = \dfrac{A}{H}$. When a vector represents the hypotenuse of a right triangle, then its magnitude is just the length of the hypotenuse so $|\vec{V}| = H$. Further, the absolute value of the $x$-component $V_x$ is just $A$, the length of the side adjacent to the angle $\theta$. The absolute value of the $y$-component $V_y$ is just $O$, the length of the side opposite $\theta$. Rearranging the formulas for $V_x$ and $V_y$ above recovers the formulas for $\sin(\theta)$ and $\cos(\theta)$ from your trigonometry class.
Apply
When we are working on real-world problems, we often do not use the standard angle, but instead discuss angles in terms of the cardinal directions (as you have seen in previous episodes). Don’t forget that north is positive in the $y$-direction and east is positive in the $x$-direction. It always helps to draw and label a diagram!
