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Chapter 19: Problem 18

Show that the volume \(V\) of the parallelepiped with adjacent edges \(\mathbf{a}, \mathbf{b}\) and \(\mathbf{c}\) is given as the magnitude of the scalar triple product. That is: $$ V=|\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c})|=|\mathbf{b} \cdot(\mathbf{c} \times \mathbf{a})|=|\mathbf{c} \cdot(\mathbf{a} \times \mathbf{b})|. $$

Short Answer

Expert verified
The volume \( V \) is given by \( V=|\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c})| \) because it represents the scalar triple product's magnitude, equivalent under cyclic permutations.

Step by step solution

01

Understand the Problem Statement

We need to show that the volume of a parallelepiped with edges represented by three vectors \( \mathbf{a} \), \( \mathbf{b} \), and \( \mathbf{c} \) can be calculated using the scalar triple product. The expression for the volume is given by \( V = |\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})| \), or equivalently by permutations of these vectors.
02

Concept of Scalar Triple Product

The scalar triple product \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \) represents the volume of the parallelepiped spanned by vectors \( \mathbf{a} \), \( \mathbf{b} \), and \( \mathbf{c} \). This result comes from the fact that \( \mathbf{b} \times \mathbf{c} \) gives a vector orthogonal to the plane containing \( \mathbf{b} \) and \( \mathbf{c} \). Multiplying by \( \mathbf{a} \) through the dot product scales this to a scalar representing the parallelepiped's volume.
03

Compute the Cross Product

Compute \( \mathbf{b} \times \mathbf{c} \). This results in a vector perpendicular to both \( \mathbf{b} \) and \( \mathbf{c} \). Let's denote this vector as \( \mathbf{n} = \mathbf{b} \times \mathbf{c} \). If vectors \( \mathbf{b} \) and \( \mathbf{c} \) are in \( \mathbb{R}^3 \), then \( \mathbf{n} = (b_2c_3 - b_3c_2, b_3c_1 - b_1c_3, b_1c_2 - b_2c_1) \).
04

Compute Dot Product with Vector \( \mathbf{a} \)

Now take the dot product of \( \mathbf{a} \) with the vector \( \mathbf{n} \) obtained from the cross-product. The expression is \( \mathbf{a} \cdot \mathbf{n} = a_1(b_2c_3 - b_3c_2) + a_2(b_3c_1 - b_1c_3) + a_3(b_1c_2 - b_2c_1) \). This is the same as the determinant of a 3x3 matrix with \( \mathbf{a} \), \( \mathbf{b} \), and \( \mathbf{c} \) as its rows.
05

Calculate the Magnitude

Finally, take the absolute value of the dot product computed in the previous step: \( V = |\mathbf{a} \cdot \mathbf{n}| \). This absolute value ensures the volume is non-negative, as volume should always be a positive quantity, regardless of the orientation of the vectors. Thus, the volume is \( V = |\det([\mathbf{a}, \mathbf{b}, \mathbf{c}])| \).
06

Confirm Symmetry in Expressions

Recognize that by vector properties, the scalar triple product is invariant under cyclic permutation, confirming the cyclic nature: \( V=|\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c})| = |\mathbf{b} \cdot(\mathbf{c} \times \mathbf{a})| = |\mathbf{c} \cdot(\mathbf{a} \times \mathbf{b})| \). This means any reordering of the vectors cyclically gives the same absolute value, confirming it all results in the same volume.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Scalar Triple Product
In vector calculus, the scalar triple product is an essential operation used to find the volume of a parallelepiped, which is a 3D figure bordered by parallelogram faces. If you're dealing with vectors \(\mathbf{a}, \mathbf{b}, \mathbf{c}\), the scalar triple product is expressed as \(\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})\). What makes this product compelling is the combination of both dot and cross products.
  • **Dot Product**: This operation between two vectors gives a scalar quantity.
  • **Cross Product**: This is an operation resulting in a vector that is perpendicular to the plane formed by the initial two vectors.
The scalar triple product results in a scalar value, which geometrically signifies the volume of the parallelepiped. Its key property is the invariance under cyclic permutations, ensuring equivalent volume measurements regardless of the vector's order in the operation.
Cross Product
The cross product is a crucial vector operation in three-dimensional space. When you compute the cross product of two vectors \(\mathbf{b}\) and \(\mathbf{c}\), you obtain a new vector, \(\mathbf{b} \times \mathbf{c}\), which is perpendicular to the plane containing \(\mathbf{b}\) and \(\mathbf{c}\).
  • This perpendicular vector has a magnitude equal to the area of the parallelogram spanned by \(\mathbf{b}\) and \(\mathbf{c}\).
  • The direction is determined by the right-hand rule, ensuring the orientation is consistent.
More mathematically, if \(\mathbf{b} = (b_1, b_2, b_3)\) and \(\mathbf{c} = (c_1, c_2, c_3)\), the resulting vector has components:\[\mathbf{b} \times \mathbf{c} = (b_2c_3 - b_3c_2, b_3c_1 - b_1c_3, b_1c_2 - b_2c_1)\]This vector is then used in the scalar triple product to calculate the volume of a parallelepiped.
Volume of Parallelepiped
Calculating the volume of a parallelepiped using vectors is a practical application of the scalar triple product. A parallelepiped is defined by its three adjacent edges, represented by vectors \(\mathbf{a}, \mathbf{b},\) and \(\mathbf{c}\).To find the volume:1. **Calculate the Cross Product**: First, compute the cross product \(\mathbf{b} \times \mathbf{c}\) to obtain a vector perpendicular to \(\mathbf{b}\) and \(\mathbf{c}\).2. **Perform the Dot Product**: Use the result from the cross product and dot it with the third vector, \(\mathbf{a}\), giving the volume of the parallelepiped.3. **Take the Absolute Value**: Since volume can鈥檛 be negative, take the absolute value, ensuring a non-negative result.The formula is structured as:\[V = |\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})|\]This method beautifully leverages vector calculus, offering an efficient route to determine the volume, and showcases the cyclical property of the scalar triple product, ensuring the volume remains unchanged under cyclic permutations of the vectors.

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Most popular questions from this chapter

Find the scalar product a.b and the vector product \(\mathbf{a} \times \mathbf{b}\), when (a) \(\mathbf{a}=\mathbf{i}+2 \mathbf{j}-\mathbf{k}, \mathbf{b}=2 \mathbf{i}+3 \mathbf{j}+\mathbf{k}\) (b) \(\mathbf{a}=2 \mathbf{i}+3 \mathbf{j}+4 \mathbf{k}, \mathbf{b}=5 \mathbf{i}-2 \mathbf{j}+\mathbf{k}\)

If position vectors, \(\overline{\mathrm{OA}}, \overline{\mathrm{OB}}, \overline{\mathrm{OC}}\), are defined by \(\overline{\mathrm{OA}}=2 \mathbf{i}-\mathbf{j}+3 \mathbf{k}\), \(\overline{\mathrm{OB}}=3 \mathbf{i}+2 \mathbf{j}-4 \mathbf{k}, \overline{\mathrm{OC}}=-\mathbf{i}+3 \mathbf{j}-2 \mathbf{k}\), determine: (a) the vector \(\overline{\mathrm{AB}}\) (b) the vector \(\overline{\mathrm{BC}}\) (c) the vector product \(\overline{\mathrm{AB}} \times \overline{\mathrm{BC}}\) (d) the unit vector perpendicular to the plane \(\mathrm{ABC}\).

If \(\mathbf{a}=2 \mathbf{i}+2 \mathbf{j}-\mathbf{k}\) and \(\mathbf{b}=3 \mathbf{i}-6 \mathbf{j}+2 \mathbf{k}\), find (a) a.b and (b) \(\mathbf{a} \times \mathbf{b}\).

If \(\mathbf{a}=3 \mathbf{i}-\mathbf{j}+2 \mathbf{k}, \mathbf{b}=\mathbf{i}+3 \mathbf{j}-2 \mathbf{k}\), determine the magnitude and direction cosines of the product vector \((\mathbf{a} \times \mathbf{b})\) and show that it is perpendicular to a vector \(\mathbf{c}=9 \mathbf{i}+2 \mathbf{j}+2 \mathbf{k}\)

Prove that the area of the parallelogram with adjacent sides \(\mathbf{a}\) and \(\mathbf{b}\) is given as \(|\mathbf{a} \times \mathbf{b}|\)
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