How to Calculate the Volume of a Parallelepiped

The volume of a parallelepiped defined by three vectors is a key concept in vector algebra, frequently examined in advanced mathematics and engineering contexts. Analysing its computation requires understanding the scalar triple product framework and geometric interpretation in three-dimensional space.

Definition of Volume of a Parallelepiped

A parallelepiped is a six-faced solid where each face is a parallelogram, constructed using three non-coplanar vectors as adjacent edges from a common vertex. Its volume quantifies the three-dimensional space enclosed by these vectors.

Mathematically, if the vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$ represent the coterminous edges of a parallelepiped, then its volume $V$ is given by the absolute value of the scalar triple product:

$V = |\vec{a}\cdot(\vec{b}\times \vec{c})|$

This formula remains fundamental in problems involving volume determination, including those found in JEE Main mathematics.

Geometric Interpretation

The magnitude of the cross product $\vec{b}\times\vec{c}$ yields the area of the parallelogram formed by $\vec{b}$ and $\vec{c}$. The dot product with $\vec{a}$ extracts the component of $\vec{a}$ perpendicular to this area, representing the height.

Therefore, the scalar triple product computes the product of base area and height, corresponding to the standard geometric volume formula. The sign may be positive or negative depending on orientation, hence the absolute value is used for volume.

This geometrical reasoning forms the basis for the vector formula's physical relevance in three-dimensional space.

Expression in Determinant Form

The scalar triple product can be calculated as the determinant of a $3\times3$ matrix whose rows, or columns, are the components of the given vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$ respectively.

For $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$, $\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$, and $\vec{c} = c_1\hat{i} + c_2\hat{j} + c_3\hat{k}$, the volume formula becomes:

$V = \left| \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix} \right|$

This determinant provides an algebraically efficient approach and can be used in computational settings, such as a volume of parallelepiped calculator.

Key Properties of the Scalar Triple Product

• The scalar triple product is invariant under cyclic permutations
• Its absolute value gives the parallelepiped’s volume
• If vectors are coplanar, the triple product equals zero
• Changing the order to an odd permutation changes the sign
• Expressible as a determinant composed of vector components
• Useful for checking vector coplanarity and calculating volumes

Example Calculation

Consider vectors $\vec{a} = 2\hat{i} - 3\hat{j} + 4\hat{k}$, $\vec{b} = \hat{i} + 2\hat{j} - \hat{k}$, and $\vec{c} = 3\hat{i} - \hat{j} + 2\hat{k}$ as coterminous edges of a parallelepiped.

The volume is given by

$V = \left| \begin{vmatrix} 2 & -3 & 4 \\ 1 & 2 & -1 \\ 3 & -1 & 2 \end{vmatrix} \right|$

Evaluating the determinant:

$\begin{aligned} V = |&2[(2)(2) - (-1)(-1)] \\ &- (-3)[(1)(2) - (-1)(3)] \\ &+ 4[(1)(-1) - (2)(3)]| \end{aligned}$

Calculate each term: $(2)(2) - (-1)(-1) = 4-1 = 3$, $(1)(2) - (-1)(3) = 2+3 = 5$, $(1)(-1)-(2)(3) = -1-6 = -7$. Thus, $V = |2\times3 + 3\times5 + 4\times(-7)| = |6 + 15 - 28| = |-7| = 7$.

Therefore, the volume of the parallelepiped is $7$ cubic units.

Derivation of Volume Formula Using Vectors

Let vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$ emanate from the same vertex. The base area is $|\vec{b}\times\vec{c}|$. The height is the projection of $\vec{a}$ onto the direction normal to the base, so $h = |\vec{a}\cdot \hat{n}|$, where $\hat{n}$ is the unit vector in the direction of $\vec{b}\times\vec{c}$.

The unit normal vector is $\hat{n} = \dfrac{\vec{b}\times\vec{c}}{|\vec{b}\times\vec{c}|}$. Therefore, $h = |\vec{a}\cdot \dfrac{\vec{b}\times\vec{c}}{|\vec{b}\times\vec{c}|}|$.

The volume is $V = $ (base area) $\times$ (height) $= |\vec{b}\times\vec{c}| \cdot |\vec{a}\cdot \dfrac{\vec{b}\times\vec{c}}{|\vec{b}\times\vec{c}|}| = |\vec{a}\cdot(\vec{b}\times\vec{c})|$.

Related Problems and Applications

The volume formula is essential for verifying vector coplanarity, since $|\vec{a}\cdot(\vec{b}\times\vec{c})|=0$ indicates coplanar vectors. It is widely applied in geometry, physics, and engineering, with further conceptual insights provided in Geometry Of Complex Numbers.

Summary Table: Parallelepiped Volume Calculation

Mastery of the parallelepiped volume formula enables systematic analysis of complex vector problems in spatial geometry. The determinant approach streamlines calculations, and its fundamental properties assist in theoretical and applied mathematics. Extensions of this idea are discussed in Acquisition Of Volume.