Vectors

The volume of a parallelepiped formed by three vectors is given by the absolute value of their scalar triple product, which is calculated as the determinant of the matrix formed by their components. We are given that the volume is 15. Let's evaluate the determinant:

So, we have the equation . This gives two possibilities:

For equation (2), the discriminant is , so it has no real solutions.

For equation (1), , we can factor it: This gives two possible values for : or . From the given options, is a valid solution.

If three vectors are coplanar (lie in the same plane), their scalar triple product is zero. The scalar triple product is calculated as the determinant of the matrix formed by the components of the vectors.

Let the vectors be , , and .

For them to be coplanar, . Expand the determinant along the second row for easier calculation: \\ Since a, b, c are non-negative numbers, we can take the square root:

This is the definition of the Geometric Mean of a and b.

We are given that and are unit vectors, and the equation is: Since and are unit vectors, we know that: Step 1: Check if is parallel to To determine if is parallel to , we examine the given equation. For the left-hand side to equal 3, the vectors must point in the same direction, which suggests that and must be parallel. If they were not parallel, the sum of the vectors would result in a different magnitude.

Thus, the equation implies that and are in the same direction. Therefore, is parallel to .

Conclusion: The correct answer is .