Episode title: Magnitude of the Munchies
Preview
DOT is still learning how to walk directly toward one of their snacks. In the previous episode, we told DOT which direction to walk in by giving them the angle of the displacement vector. In this episode, we’ll tell DOT how far to walk to pick up a snack. To determine how far DOT should walk, we adjust the grid that DOT is walking on. The new grid shows us polar coordinates, which indicate direction and distance.
Play
[DOT game still in development!]
Practice
In the previous episode, we learned how to determine the standard angle $\theta_V$ of a vector $\vec{V}$ when it is given in rectangular components $\langle V_x, V_y \rangle$. Now we determine the length (sometimes called magnitude) $|\vec{V}|$ of the vector $\vec{V}$. Together, $\theta_V$ and $|\vec{V}|$ are called the polar coordinates of the vector $\vec{V}$.
Notice that whether we are using rectangular form or polar form, we always need two pieces of information to describe a vector $\vec{V}$:
- Rectangular form: $V_x$ and $V_y$,
- Polar form: $\theta_V$ and $|\vec{V}|$.
If we know $V_x$ and $V_y$, determining the length $\vec{V}$ is straightforward:
$|\vec{V}| = \sqrt{(V_x)^2+(V_y)^2}$
You may have seen formula for the length of a vector before. It’s just the formula for distance between two points in the plane. You might also recognize it as a version of the Pythagorean theorem, which gives the length of the hypotenuse of a right triangle in terms of the lengths of the other two sides. Here, the vector itself plays the role of the hypotenuse while the $x$-component and $y$-component play the roles of the other two sides (it doesn’t matter if $V_x$ or $V_y$ are positive or negative since in the formula we are squaring them).
