Find the volume of the parallelepiped having u = 3i - 5j + k , v= 2j and w= 3i + j + k as adjacent edges.
the volume is just |u•v×w|
That's pretty straightforward. What do you get?
To find the volume of a parallelepiped, we can use the scalar triple product of the vectors representing its adjacent edges.
The scalar triple product is given by:
V = u · (v × w)
First, let's find the cross product of v and w:
v × w = (2j) × (3i + j + k)
= 2j × 3i + 2j × j + 2j × k
= 6ij + 0 + 2kj
= 6ij + 2kj
Next, let's find the dot product of u and the cross product of v and w:
u · (v × w) = (3i - 5j + k) · (6ij + 2kj)
= 3(6i · i) + (-5j · 6ij) + (k · 2kj)
= 18i · i - 30ij · j + k · 2k
= 18i · i - 30 + 2k · k
= 18(1) - 30 + 2(1)
= 18 - 30 + 2
= -10
Therefore, the volume of the parallelepiped is -10 cubic units.
To find the volume of a parallelepiped, you can use the scalar triple product of the vectors representing the adjacent edges.
First, let's calculate the scalar triple product. The scalar triple product of three vectors u, v, and w is given by the dot product of the vector u with the cross product of vectors v and w:
Scalar triple product (u · (v × w))
To calculate the cross product of vectors v and w, perform the following calculations:
v × w = (2j) × (3i + j + k)
= (2j × 3i) + (2j × j) + (2j × k)
= 6(i × j) + 0 + 2j × k
Now, let's calculate i × j:
i × j = k
Substituting the cross products back into the equation:
v × w = 6k + 0 + 2j × k
= 6k + 0 + 2(-i)
= -2i + 6k
Thus,
u · (v × w) = (3i - 5j + k) · (-2i + 6k)
= -6(i · i) + 18(k · i) - 10(j · i) + 3(i · k) - 15(j · k) + 5(k · k)
= -6(1) + 18(0) - 10(0) + 3(0) - 15(0) + 5(1)
= -6 + 5
= -1
The volume of the parallelepiped is the absolute value of the scalar triple product.
Volume of parallelepiped = |u · (v × w)|
= |-1|
= 1
Therefore, the volume of the parallelepiped is 1.