What is the formula for volume of a parallelepiped?
The volume of the parallelepiped spanned by a, b, and c is Volume=area of base⋅height=∥a×b∥ ∥c∥ |cosϕ|=|(a×b)⋅c|. The formula results from properties of the cross product: the area of the parallelogram base is ∥a×b∥ and the vector a×b is perpendicular to the base. The height of the parallelepiped is ∥c∥ |cosϕ|.
Is a tetrahedron a parallelepiped?
Prism is a 3D shape with two equal polygonal bases whose corresponding vertices can be (and are) joined by parallel segments. Parallelepiped is a prism with parallelogram bases. A tetrahedron is isosceles iff the associated parallelepiped is a cuboid.
Why is volume of parallelepiped?
The height of the parallelepiped is the component of c in the direction normal to the base, i.e., in the direction of a×b. Hence the height is ∥c∥ |cosϕ|, where ϕ is the angle between c and a×b. The volume of the parallelepiped is therefore Volume=∥a×b∥ ∥c∥ |cosϕ|=|(a×b)⋅c|.
How do you find the volume of a tetrahedron given the vertices?
A tetrahedron has four faces, six edges and four vertices. Its three edges meet at each vertex. The four vertices that we have been given in the question are A(1,1,0) B(-4,3,6) C(-1,0,3) and D(2,4,-5). Hence, the volume of the given tetrahedron is 6 cubic units.
Can volume of a tetrahedron be zero?
If you start with an equilateral triangle, the result is a regular tetrahedron. If you start with an acute triangle, the result is an equifacial tetrahedron. If you start with an right triangle, the result is a “degenerate” tetrahedron with zero volume.
What is centroid of tetrahedron?
The centroid of a tetrahedron is the midpoint between its Monge point and circumcenter. These points define the Euler line of the tetrahedron that is analogous to the Euler line of a triangle.
How do you solve parallelepiped?
The volume of the parallelepiped is therefore Volume=∥a×b∥ ∥c∥ |cosϕ|=|(a×b)⋅c|. (Remember the definition of the dot product.) Using the formula for the cross product in component form, we can write the scalar triple product in component form as (a×b)⋅c=|a2a3b2b3|c1−|a1a3b1b3|c2+|a1a2b1b2|c3=|c1c2c3a1a2a3b1b2b3|.