Question

What is the reason that the sum of the magnitudes of two vectors and the magnitude of the sum of two vectors not equal? Thank you in advanced

Answer 1

Explanation:

It is the same reason that the distance by road is not the same as the distance "as the crow flies." The two vectors are often not aligned so that the magnitudes both add to directly to the distance from the origin (or the tail of the first vector).

For example, suppose you walk two segments of 1 mile each. If you walk east in both cases, you end up 2 miles east of where you started. (The sum of the vectors is the sum of their magnitudes.)

If you walk east 1 mile and north 1 mile, you end up about 1.4 miles from where you started, not 2 miles. The second "vector" did not add directly to the distance from your starting point.

If you walk east 1 mile, then west 1 mile, you end up exactly where you started. The sum of the vectors is zero, but the sum of their magnitudes is still 2 miles.