In physics, sometimes you have to find the angle and magnitude of a vector rather than the components. To find the magnitude, you use the Pythagorean theorem. And to find

you use the inverse tangent function (or inverse sine or cosine).

For example, assume you’re looking for a hotel that’s 20 miles due east and then 20 miles due north. From your present location, what is the angle (measured from east) of the direction to the hotel, and how far away is the hotel? You can write this problem in vector notation, like so:

Step 1: (20, 0)

Step 2: (0, 20)

When adding these vectors together, you get this result:

(20, 0) + (0, 20) = (20, 20)

The resultant vector is (20, 20). That’s one way of specifying a vector — use its components. But this problem isn’t asking for the results in terms of components. The question wants to know the angle and distance to the hotel.

Using the angle created by a vector to get to a hotel.

In other words, looking at the above figure, the problem asks,

If you know a vector’s vertical and horizontal components, finding the vector’s magnitude isn’t so hard because you just need to find the hypotenuse of a right triangle. You can use the Pythagorean theorem (x2 + y2 = h2), solved for h:

Plugging in the numbers gives you

Keep in mind that when you know the horizontal and vertical components of a vector, you can use the tangent to find the angle because

All you have to do is take the inverse tangent of y/x:

Suppose you drive 20 miles east and 20 miles north. Here’s how you find