Can a vector be broken down to its components?

Question

Math question. Hi. Here is the URL with some examples: http://hyperphysics.phy-astr.gsu.edu/hbase/vect.html#vec9 , bottom of the page.
Suppose we have four vectors (A, B, C, D). We know their magnitudes and angles (directions). We add them and produce a new vector R. Let's consider a different situation. Let's say we are given only the resulting vector R (magitude and angle) and number of vectors (4) added to generate the R vector. Can we determine magnitudes and angles of these individual vectors (all of which are unknown)? Thanks.

Answer 1

No, you can't. Imagine we keep vector R the same, but draw another series of vectors A, B, C, D that terminate in the same place. That becomes another vector sum, which means decomposition of R isn't unique.

Answer 2

To find the vectors A, B, C and D in the diagram, you need to measure the length of each one and the angle each one makes with the horizontal. Fill the magnitudes in the left hand column and the angles in the right hand column of the table.

To get the x-components, multiply the length of each vector by the cosine of its angle.
To get the y-components, multiply the length of each vector by the sine of its angle.
Add the four quantities in each case to get the x- and y- components of R.

You could not work out the four vectors if you had no information other than the magnitude and direction of R, because any four vectors placed end to end starting where R starts and ending where R ends would yield the same result.

Answer 3

no. there are infinitely man solutions. In fact, pick any vector for A, B, and C. Then let D = R - A -B -C. and A,B,C,D is a solution.