Final answer:
To determine which vector does not belong to the subspace spanned by -2i + 3j+k and i + j + 2k, we can set up a system of equations to check. After checking each vector, we find that the vector -41 + 6j + 2k does not belong to the subspace.
Step-by-step explanation:
To determine which vector does not belong to the subspace spanned by -2i + 3j+k and i + j + 2k, we need to check if each vector can be written as a linear combination of these two vectors. We can set up a system of equations to check:
a) -41 + 6j + 2k:
Multiplying the first vector by a constant, say x, and the second vector by a constant, say y, and then adding them should result in -41 + 6j + 2k. However, there are no values of x and y that satisfy this equation, so this vector does not belong to the subspace.
b) 3i + 8j + 11k:
Multiplying the first vector by a constant, say x, and the second vector by a constant, say y, and then adding them should result in 3i + 8j + 11k. There are values of x and y that satisfy this equation, so this vector belongs to the subspace.
c) 111 - 4j + 6k:
Multiplying the first vector by a constant, say x, and the second vector by a constant, say y, and then adding them should result in 111 - 4j + 6k. There are values of x and y that satisfy this equation, so this vector belongs to the subspace.
d) 101 - 5j + 5k:
Multiplying the first vector by a constant, say x, and the second vector by a constant, say y, and then adding them should result in 101 - 5j + 5k. There are values of x and y that satisfy this equation, so this vector belongs to the subspace.
Therefore, the vector that does not belong to the subspace is a) -41 + 6j + 2k.