Calculations with Magnitude and Direction

If you ask someone where Las Vegas is compared to Los Angeles, they are unlikely to say that the Δx is 290 km and the Δy is 230 km, in a coordinate system where the positive x axis is east and the y axis points north. They will probably say instead that it's 370 km to the northeast. If they were being precise, they might specify the direction as 38 ° counterclockwise from east. In two dimensions, we can always specify a vector's direction like this, using a single angle. A magnitude plus an angle suffice to specify everything about the vector. The following two examples show how we use trigonometry and the Pythagorean theorem to go back and forth between the x-y and magnitude-angle descriptions of vectors.

Finding the magnitude and angle from the components.

Finding the components from the magnitude and angle.

The following example shows the correct handling of the plus and minus signs, which is usually the main cause of mistakes.

Negative components.

Discussion Question

A	In the example above, we dealt with components that were negative. Does it make sense to talk about positive and negative vectors?