Final answer:
The volume of the parallelepiped formed by the vectors u, v, and w is found using the triple scalar product. The volume is zero in this case, indicating that the vectors are co-planar and do not form a three-dimensional shape.
Step-by-step explanation:
To find the volume of the parallelepiped formed by vectors u, v, and w, we can use the triple scalar product. In this case, the vectors given are u = i + j, v = j + k, and w = i + k. The triple scalar product is calculated by taking the dot product of one of the vectors with the cross product of the other two vectors. The volume V can be found by:
V = |u · (v × w)|
First, we calculate the cross product v × w:
- v × w = (j + k) × (i + k)
- = j × i + j × k + k × i + k × k
- = -i + (k × i) + (j × k) + 0
- = -i + k - j
Next, we take the dot product of u with the result:
- u · (-i + k - j) = (i + j) · (-i + k - j)
- = i · (-i) + i · k + i · (-j) + j · (-i) + j · k + j · (-j)
- = 0 + 0 + 0 + 0 + 0 + 0
- = 0
The volume of the parallelepiped is therefore 0, indicating that the vectors are co-planar and do not form a three-dimensional parallelepiped.