Problem 24 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the volume of the parallelepiped with edges \(3 \mathbf{i}-4 \mathbf{j}+2 \mathbf{k},-\mathbf{i}+2 \mathbf{j}+\mathbf{k}\), and \(3 \mathbf{i}-2 \mathbf{j}+5 \mathbf{k}\)

Step-by-Step Solution

Answer

The volume of the parallelepiped is 12 cubic units.

Step 1: Understanding the Parallelepiped

A parallelepiped is a six-faced figure (also called a polyhedron) where each face is a parallelogram. The volume of the parallelepiped spanned by vectors \( \mathbf{a} \), \( \mathbf{b} \), and \( \mathbf{c} \) is given by the scalar triple product \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \). The vectors given are \( \mathbf{a} = 3\mathbf{i} - 4\mathbf{j} + 2\mathbf{k} \), \( \mathbf{b} = -\mathbf{i} + 2\mathbf{j} + \mathbf{k} \), and \( \mathbf{c} = 3\mathbf{i} - 2\mathbf{j} + 5\mathbf{k} \).

Step 2: Calculate the Cross Product \( \mathbf{b} \times \mathbf{c} \)

The cross product \( \mathbf{b} \times \mathbf{c} \) is calculated using the determinant of a matrix with the unit vectors \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) in the first row, the components of \( \mathbf{b} \) in the second row, and the components of \( \mathbf{c} \) in the third row: \[ \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ -1 & 2 & 1 \ 3 & -2 & 5 \end{vmatrix}\]This evaluates to: \(- 4\mathbf{i} - 8\mathbf{j} - 4\mathbf{k}\).

Step 3: Calculate the Scalar Triple Product

The scalar triple product \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \) is a dot product of \( \mathbf{a} = 3\mathbf{i} - 4\mathbf{j} + 2\mathbf{k} \) with the cross product result \(-4\mathbf{i} - 8\mathbf{j} - 4\mathbf{k}\). Calculate as follows: \[ (3\mathbf{i} - 4\mathbf{j} + 2\mathbf{k}) \cdot (-4\mathbf{i} - 8\mathbf{j} - 4\mathbf{k}) = 3(-4) + (-4)(-8) + 2(-4) \]This equals: \(-12 + 32 - 8 = 12\).

Step 4: Result Interpretation

The absolute value of the scalar triple product gives the volume of the parallelepiped. Therefore, the volume is \(|12| = 12\).

Key Concepts

Parallelepiped Volume

A parallelepiped is a three-dimensional geometric figure commonly seen in geometry problems. It resembles a skewed box, where each of the six faces is a parallelogram. The volume of a parallelepiped plays a crucial role in understanding the space it occupies.
To determine its volume, you need three vectors that represent its edges that meet at a single vertex. The essential formula used for volume calculation involves a mathematical operation known as the scalar triple product.
The scalar triple product is a clever algebraic tool that combines three vectors to yield a scalar, representing the volume in this context. For vectors \( \mathbf{a} \), \( \mathbf{b} \), and \( \mathbf{c} \), the volume is given by \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \). Essentially, this expression combines vector multiplication to capture the three-dimensional spread of the parallelepiped.

Scalar Triple Product

The scalar triple product is a highly useful operation in vector calculus used not only for finding volumes but also in various applications involving torque and angular momentum. It is expressed as \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \). This expands into a sequence of vector operations:

• The first operation is a cross product, \( \mathbf{b} \times \mathbf{c} \), resulting in a vector. This vector is perpendicular to both \( \mathbf{b} \) and \( \mathbf{c} \), following the right-hand rule common in vector math.
• The second operation is a dot product, where vector \( \mathbf{a} \) is dotted with the result of the cross product. This reduces the perpendicular vector to a scalar, indicating how much \( \mathbf{a} \) aligns with \( \mathbf{b} \times \mathbf{c} \).

Interestingly, the scalar triple product can be positive or negative. The magnitude of this result—the absolute value—directly reflects the volume of the parallelepiped. It shows how three vectors span three-dimensional space.

Cross Product Calculation

The cross product is fundamental to vector calculus. It is a binary operation on two vectors in three-dimensional space and results in another vector. This new vector is orthogonal to the plane formed by the original two vectors.
To compute the cross product \( \mathbf{b} \times \mathbf{c} \), a determinant of a 3x3 matrix is calculated. The matrix consists of:

• The unit vectors \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) making up the first row, which are the standard basis vectors in 3D space.
• The components of vector \( \mathbf{b} \) fill the second row.
• The components of vector \( \mathbf{c} \) populate the third row.

By calculating this determinant, we acquire the vector orthogonal to both \( \mathbf{b} \) and \( \mathbf{c} \). In our exercise, the cross product result is \(-4\mathbf{i} - 8\mathbf{j} - 4\mathbf{k}\), illustrating the direction and magnitude of perpendicularity relative to the original vectors. This intermediate step is vital for determining spatial relationships and features such as volume.