Vectors

Step 1: Use the formula for the magnitude of the sum of two vectors.
The magnitude of is given by: Since both and are unit vectors, , so:
Step 2: Set the magnitude equal to 1 for it to be a unit vector.
For to be a unit vector, its magnitude must be 1: Squaring both sides:
Step 3: Conclusion.
This equation holds true when , corresponding to option (A).

Step 1: Recall the formula for the volume of the parallelepiped.
The volume of a parallelepiped formed by three vectors is given by the scalar triple product:
Step 2: Compute the cross product .
Let , . First, compute the cross product . Then, take the dot product of with this result.
Step 3: Find the result of the scalar triple product.
After calculating, we get the volume as 8 cubic units.

Step 4: Conclusion.
Thus, the volume of the parallelepiped is 8, which corresponds to option (A).