Vectors II:

More about vectors. The whole idea is this: Since any vector can be represented by any other combination of vectors that starts and ends where the initial one does, we can replace any vector by "orthogonal" vectors along the x-, y-, and z-axes. While this gives us more vectors, they are orthogonal and so can be treated independently. In fact, this approach is so useful that we try to always write vectors this way. (Remembering that if we should ever need to write them as an arrow at some angle we can use the Pythagorean theorem and definition of the tangent of an angle to do this). To help implement this approach we invent "unit vectors", one for each coordinate axis. As the name implies, a unit vector has length 1 and points along the x-, y- or z-axes. Here's an example:

In the drawing the vector A is 10 units long (10 feet, 10 meters/second or whatever) and points at an angle 30 degrees (always measured from the x-axis, positive in the counter-clockwise direction). Its x-component is 10 cos(30) = 8.66, and its y-component is 10 sin(30) = 5. The unit vectors point along the x- and y-axes. In terms of the unit vectors, the vector A can be written as The unit vector notation (it is, after all, just a notation - a shorthand way of writing something down) separates a vector's magnitude from its orientation. Any vector can have its length changed by multiplying it by a scalar, so you can think of the components of the vector A as having their unit vector magnitudes multiplied by the scalars 8.66 and 5. One immediate advantage of this is to allow us to add and subtract vectors easily. Example: Add the vectors The result is simply We will seldom use vectors in any other form than that of unit vectors from now on.