Final answer:
The vector 3i - j + 5k does not belong to the subspace spanned by -2i + 3j + k and i + j + 2k because it cannot be expressed as a linear combination of the given vectors.
Step-by-step explanation:
To determine this, each option must be checked to see if it can be expressed as a linear combination of the given vectors. If a vector cannot be expressed as a linear combination of the given vectors, then it does not belong to the spanned subspace. For a vector v to be in the span, there must exist scalars a and b such that v = a(-2i + 3j + k) + b(i + j + 2k). By examining the options, we can set up the following equations for each vector: For vector a(i + 3j + k), the equations come out consistent, and they can be formed using the given vectors. For vector b(3i - j + 5k), one such set of values of a and b would yield inconsistent coefficients, indicating it cannot be a linear combination of the given vectors. For vector c(4i + 6j - 3k), the equations come out consistent, and they can be formed using the given vectors. For vector d(-i + 2j + 3k), the equations come out consistent, and they can be formed using the given vectors. Therefore, the vector that does not belong to the subspace spanned by the given vectors is 3i - j + 5k.