Final answer:
Using the vector projection of s on r along with the given angle between the vectors and the ratio of their magnitudes, we can determine the magnitude and direction of vector r through a series of calculations involving dot products and scalar multiplication.
Step-by-step explanation:
The projection of vector s on vector r, given as (2,-1,7), can be used to determine the magnitude and direction of vector r. The projection formula states that the vector projection of s on r is equal to ((s · r)/|r|^2) * r, where '·' denotes the dot product and |r| denotes the magnitude of r. Knowing that vector s is half the length of vector r and the angle between them is 120°, the magnitude of vector r can be determined using the dot product and the given vectors.
The dot product s · r equals the magnitude of the projection times the magnitude of r (because the projection of s on r is along the direction of r). Since the angle between vector s and vector r is 120°, the cosine of this angle is -1/2, which influences the dot product; hence, the projection scalar equals |s| * |r| * cos(120°). Finally, substitute the known values into the formula and solve for |r| to find vector r with the correct magnitude, then apply scalar multiplication to the unit vector in the direction of r to get the final vector.