How to Calculate the Volume of a Parallelepiped

A parallelepiped is a three-dimensional solid whose faces are parallelograms. This geometric structure arises naturally when three distinct vectors are positioned such that they originate from a common point in space. The intersection of these vectors establishes the shape and orientation of the parallelepiped within a given coordinate system.

In coordinate geometry and vector algebra, determining the volume of a parallelepiped whose edges align with given vectors is of fundamental significance. This calculation relies on understanding vector products and their geometric interpretation. The mathematical formulation aligns closely with concepts from scalar triple product theory, as the spatial relation among vectors sets the volumetric capacity.

Formal Definition and Geometric Structure

A parallelepiped is defined as a solid bounded by six parallelogram faces. Each pair of opposite faces is parallel, and the solid has eight vertices and twelve edges. The edges that meet at one vertex can be represented by three non-coplanar vectors, usually denoted as $\vec{a}$, $\vec{b}$, and $\vec{c}$.

The position vectors of the vertices originating from a chosen origin yield the edge vectors of the parallelepiped. The geometric figure is entirely determined by these three vectors provided they are not linearly dependent.

Volume of a Parallelepiped: Vector Formula

The volume $V$ of a parallelepiped whose coterminous edges are represented by vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$ is given by the absolute value of the scalar triple product:

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

Here, the operation $\vec{b} \times \vec{c}$ yields a vector orthogonal to the plane containing $\vec{b}$ and $\vec{c}$, with magnitude equal to the area of the parallelogram defined by these vectors. The subsequent dot product with $\vec{a}$ projects this area onto the direction of $\vec{a}$, measuring the 'height' and thus determining the solid's volume.

In terms of components, if

$ \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},\ \vec{c} = c_1\hat{i} + c_2\hat{j} + c_3\hat{k} $,

then the volume can be expressed as the absolute value of the determinant:

$ 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| $

Properties of the Volume Formula

• Zero volume if vectors are coplanar
• Non-negative due to absolute value
• Invariant under cyclic order of vectors
• Dependent on lengths and mutual angles

The volume vanishes when any two vectors are collinear or when all three lie in the same plane. This property confirms the non-degeneracy of the parallelepiped.

Component Expansion and Determinant Evaluation

To evaluate the determinant for three vectors in component form, consider:

$ \vec{a} \cdot (\vec{b} \times \vec{c}) = \begin{vmatrix} a_1 & a_2 & a_3\\ b_1 & b_2 & b_3\\ c_1 & c_2 & c_3 \end{vmatrix} $

Expand this determinant using the first row:

$ = a_1 \begin{vmatrix} b_2 & b_3\\ c_2 & c_3 \end{vmatrix} - a_2 \begin{vmatrix} b_1 & b_3\\ c_1 & c_3 \end{vmatrix} + a_3 \begin{vmatrix} b_1 & b_2\\ c_1 & c_2 \end{vmatrix} $

Each minor determinant corresponds to a pair of coordinates, capturing the oriented volume effect along the axes.

Special Case: Rectangular Parallelepiped

If the three edge vectors are mutually perpendicular and aligned with the axes, the parallelepiped forms a rectangular box. In this instance, the formula reduces to $V = |a_1b_2c_3|$, representing the product of the edge lengths along the three axes.

This case yields the familiar rectangular volume formula, in agreement with the general determinant expression.

Solved Example: Direct Determinant Evaluation

Given the vectors $\vec{a} = 2\hat{i} - 3\hat{j} + 4\hat{k}$, $\vec{b} = \hat{i} + 2\hat{j} - \hat{k}$, and $\vec{c} = 3\hat{i} - \hat{j} + 2\hat{k}$, calculate the volume of the parallelepiped formed.

First, write the determinant with the given components:

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

Expand along the first row:

$ = 2 \times \begin{vmatrix} 2 & -1 \\ -1 & 2 \end{vmatrix} - (-3) \times \begin{vmatrix} 1 & -1 \\ 3 & 2 \end{vmatrix} + 4 \times \begin{vmatrix} 1 & 2 \\ 3 & -1 \end{vmatrix} $

Compute each minor:

$ \begin{vmatrix} 2 & -1 \\ -1 & 2 \end{vmatrix} = (2)(2) - (-1)(-1) = 4 - 1 = 3 $

$ \begin{vmatrix} 1 & -1 \\ 3 & 2 \end{vmatrix} = (1)(2) - (3)(-1) = 2 + 3 = 5 $

$ \begin{vmatrix} 1 & 2 \\ 3 & -1 \end{vmatrix} = (1)(-1) - (3)(2) = -1 - 6 = -7 $

Substitute these values back:

$ V = 2 \times 3 + 3 \times 5 + 4 \times (-7) $

$ = 6 + 15 - 28 = -7 $

The final volume is $| -7 | = 7$. The magnitude ensures the volume is non-negative.

Solved Example: Coplanarity and Zero Volume

Given $\vec{a} = \hat{i} + 2\hat{j} - \hat{k}$, $\vec{b} = 2\hat{i} + 4\hat{j} - 2\hat{k}$, $\vec{c} = 3\hat{i} + 6\hat{j} - 3\hat{k}$, determine the volume of the parallelepiped.

Construct the determinant:

$ V = \left| \begin{array}{ccc} 1 & 2 & -1 \\ 2 & 4 & -2 \\ 3 & 6 & -3 \\ \end{array} \right| $

Observe that each row is a scalar multiple of the first. Thus, the vectors are coplanar, so $V = 0$.

Solved Example: Geometric Application

For a parallelepiped with edges $\vec{a}$, $\vec{b}$, $\vec{c}$ where $|\vec{a}| = 4$, $|\vec{b}| = 3$, $|\vec{c}| = 2$, and each pair is mutually perpendicular, calculate the volume.

Since the vectors are perpendicular, standard volume formula $V = |\vec{a}| \cdot |\vec{b}| \cdot |\vec{c}|$ applies:

$V = 4 \times 3 \times 2 = 24$

This aligns with the determinant formula, where the determinant is diagonal.

Common Misconceptions Regarding Volume

• Omitting absolute value can yield negative result
• Confusing order in scalar triple product
• Assuming coplanar vectors provide volume
• Misapplication for non-coterminous vectors

Typical JEE Question Patterns

• Direct computation using three vectors
• Proof of coplanarity (zero volume cases)
• Volume in terms of area and height
• Special configurations: rectangular and rhomboidal

For broader vector geometry applications, concepts such as the Area Of Triangle Formula and the Coordinate Geometry of points may be useful for related problems.