3vectors of magnitude a,2a,3a meet in a point&their directions r alongthediagonals of3adjcent faces of a cubes

Question

Determine their resultant R and its direction cosines.Also prove that sum of the 3 vectors determined by the diagonal of three adjacent faces of a cube passing thru the same corner the vector being directed from the corner is twice the vector determined by the diagonal of the cude.

Answer 1

Part One. Let
A = a sqrt(2) / 2 (magnitude a)
B = a sqrt(2) (magnitude 2a)
C = 3a sqrt(2) / 2 (magnitude 3a)

The resultant is

R = a sqrt(2) <(3/2)i + 2j + (5/2)k> (magnitude 5a)

Direction cosines:

cos alpha = 3 sqrt(2) / 10 = 0.424 (64.9 degrees)
cos beta = 2 sqrt(2) / 5 = 0.566 (55.6 degrees)
cos gamma = sqrt(2) / 2 = 0.707 (45 degrees)


Part Two. A = ; B = ; C =

Diagonal of cube D =

A + B + C = <2i + 2j + 2k> = 2D