Problem 23 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the volume of the parallelepiped with edges \(\langle 2,3,4\rangle,\langle 0,4,-1\rangle\), and \(\langle 5,1,3\rangle\) (see Example 4).

Step-by-Step Solution

Answer

The volume of the parallelepiped is 69 cubic units.

Step 1: Understand the Vectors

We are given three vectors that represent edges of a parallelepiped: \( \mathbf{a} = \langle 2, 3, 4 \rangle \), \( \mathbf{b} = \langle 0, 4, -1 \rangle \), and \( \mathbf{c} = \langle 5, 1, 3 \rangle \). To find the volume of this parallelepiped, we will use the scalar triple product.

Step 2: Calculate the Cross Product

First, find the cross-product of vectors \( \mathbf{b} \) and \( \mathbf{c} \), expressed as \( \mathbf{b} \times \mathbf{c} \). The formula for the cross-product is:\[\mathbf{b} \times \mathbf{c} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ 0 & 4 & -1 \ 5 & 1 & 3 \end{vmatrix} \]Calculate the determinant by expanding using cofactor expansion. Solving this gives:\[\mathbf{b} \times \mathbf{c} = \langle (4 \cdot 3 - (-1) \cdot 1), - (0 \cdot 3 - (-1) \cdot 5), (0 \cdot 1 - 4 \cdot 5) \rangle = \langle 13, -5, -20 \rangle\]

Step 3: Calculate the Scalar Triple Product

Next, find the scalar triple product \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \). Use the dot product formula:\[\mathbf{a} \cdot \mathbf{d} = a_1d_1 + a_2d_2 + a_3d_3\]where \( \mathbf{d} = \langle 13, -5, -20 \rangle \). Hence:\[\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = 2 \cdot 13 + 3 \cdot (-5) + 4 \cdot (-20)\]Calculate:\[26 - 15 - 80 = -69\]

Step 4: Find the Volume

The volume of the parallelepiped is the absolute value of the scalar triple product. So, the volume is:\[| -69 | = 69\] cubic units.

Key Concepts

Parallelepiped Volume

A parallelepiped is a six-faced geometrical figure, where each face is a parallelogram. The volume of this figure can be calculated if you know vectors representing three of its edges. These vectors should originate from the same vertex. Finding the volume involves understanding how these vectors interact in space using vector calculus methods.

Here's a quick way to visualize it:

• The first vector gives the length and direction of one edge.
• The second vector defines a second edge starting from the same point.
• The third does the same for the third edge.

Once you have these vectors, you can use the scalar triple product to compute the exact volume of the parallelepiped. This method is more accurate and elegant than using basic geometry since it fully captures the 3D positioning of the vectors in space.

Scalar Triple Product

The scalar triple product is an essential concept in vector calculus and plays a key role in calculating the volume of a parallelepiped. Mathematically, the scalar triple product is the dot product of one vector with the cross product of the other two vectors. This can be denoted as \[ \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \].

The result of this calculation is a scalar value.

• If the vectors are linearly dependent (i.e., they lie in the same plane), the scalar triple product is zero.
• The absolute value of the scalar triple product gives the volume of the parallelepiped formed by the vectors.

Using the scalar triple product simplifies things because it combines the dot and cross products into a single step, making it efficient when working with three vectors.

Cross Product

The cross product is a vector operation that takes two vectors and returns a third vector. This third vector is perpendicular to the plane formed by the first two, making it a useful tool for 3-dimensional calculations. The formula for the cross product of two vectors, \( \mathbf{b} = \langle b_1, b_2, b_3 \rangle \) and \( \mathbf{c} = \langle c_1, c_2, c_3 \rangle \), is:\[\mathbf{b} \times \mathbf{c} = \langle b_2c_3 - b_3c_2, b_3c_1 - b_1c_3, b_1c_2 - b_2c_1 \rangle\].

Here’s why the cross product is critical:

• It helps find a vector orthogonal to the original two, useful in physics and engineering.
• The magnitude gives the area of the parallelogram spanned by the vectors, a step towards finding the parallelepiped’s volume.

Using the cross product is direct when tackling spatial problems, especially when calculating angles or orientation.

Dot Product

The dot product, also known as the scalar product, is an operation that takes two vectors and returns a single number. This operation involves multiplying the corresponding components of each vector and summing those products. For vectors \( \mathbf{a} = \langle a_1, a_2, a_3 \rangle \) and \( \mathbf{b} = \langle b_1, b_2, b_3 \rangle \), it is expressed as:\[\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3\].

Important points about the dot product include:

• If the result is zero, the vectors are perpendicular.
• If the vectors point in exactly the same or opposite directions, it corresponds to the maximum or minimum value possible.

This operation is fundamental in finding projections of vectors and forms the final step in calculating the scalar triple product.

Vectors

Vectors are essential mathematical entities in physics and engineering. They represent quantities with both magnitude and direction, such as force, velocity, and displacement. Vectors are typically written as \( \langle x, y, z \rangle \) in 3-dimensional space and can be visualized as arrows pointing in a specific direction.

Key features of vectors include:

• Magnitude: Calculated as \( \sqrt{x^2 + y^2 + z^2} \), representing the vector's length.
• Direction: Given by the vector’s orientation in space.

Understanding vectors is crucial when studying vector calculus as they form the foundational elements for operations like the dot product and cross product, leading to calculations like the scalar triple product.