Problem 36 - Essential Calculus Early Transcendentals

Question

Find the volume of the parallelepiped with adjacent edges \(P Q, P R,\) and \(P S .\) $$P(3,0,1), \quad Q(-1,2,5), \quad R(5,1,-1), \quad S(0,4,2)$$

Step-by-Step Solution

Answer

The volume of the parallelepiped is 16 cubic units.

Step 1: Find Vectors for Adjacent Edges

First, calculate the vectors representing the edges \(PQ\), \(PR\), and \(PS\). The vectors are found by subtracting the coordinates of point \(P\) from the coordinates of points \(Q\), \(R\), and \(S\).\\[\overrightarrow{PQ} = Q - P = (-1, 2, 5) - (3, 0, 1) = (-4, 2, 4)\] \\[\overrightarrow{PR} = R - P = (5, 1, -1) - (3, 0, 1) = (2, 1, -2)\] \\[\overrightarrow{PS} = S - P = (0, 4, 2) - (3, 0, 1) = (-3, 4, 1)\]

Step 2: Calculate the Cross Product of Two Vectors

Next, find the cross product of vectors \(\overrightarrow{PR}\) and \(\overrightarrow{PS}\). The cross product \(\overrightarrow{PR} \times \overrightarrow{PS}\) will give a vector that is perpendicular to both vectors.\\[\overrightarrow{PR} \times \overrightarrow{PS} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ 2 & 1 & -2 \ -3 & 4 & 1 \end{vmatrix}\] \Calculate the determinant: \\[\overrightarrow{PR} \times \overrightarrow{PS} = \hat{i}(1\cdot1 - (-2)\cdot4) - \hat{j}(2\cdot1 - (-2)\cdot(-3)) + \hat{k}(2\cdot4 - 1\cdot(-3))\]\\[= \hat{i}(1 + 8) - \hat{j}(2 - 6) + \hat{k}(8 + 3)\] \\[= \hat{i}(9) - \hat{j}(-4) + \hat{k}(11)\] \\[= (9, 4, 11)\]

Step 3: Calculate the Volume using Scalar Triple Product

Finally, use the scalar triple product to find the volume of the parallelepiped. The volume is \(|\overrightarrow{PQ} \cdot (\overrightarrow{PR} \times \overrightarrow{PS})|\).\\[\overrightarrow{PQ} \cdot (\overrightarrow{PR} \times \overrightarrow{PS}) = \begin{bmatrix} -4 & 2 & 4 \end{bmatrix} \cdot \begin{bmatrix} 9 & 4 & 11 \end{bmatrix}\] \Calculate the dot product: \\[= (-4 \cdot 9) + (2 \cdot 4) + (4 \cdot 11)\]\\[= (-36) + 8 + 44\] \\[= 16\] \Since volume is always positive, \(|16| = 16\).

Step 4: Conclusion

The volume of the parallelepiped is 16 cubic units. The calculation involved finding vectors for the edges, computing a cross product for two of these vectors, and then finding the scalar triple product.

Key Concepts

Cross Product

In vector calculus, the cross product is a binary operation on two vectors in three-dimensional space. It results in a vector that is perpendicular to both input vectors, providing a way to calculate the area spanned by the two vectors.
For two given vectors \(\mathbf{a} = (a_1, a_2, a_3)\) and \(\mathbf{b} = (b_1, b_2, b_3)\), the cross product \(\mathbf{a} \times \mathbf{b}\) can be determined using the determinant of a matrix formed with unit vectors and the components of \(\mathbf{a}\) and \(\mathbf{b}\).

• The unit vectors \(\hat{i}, \hat{j}, \hat{k}\) are aligned with the x, y, and z axes, respectively.
• The formula for the cross product is: \[\mathbf{a} \times \mathbf{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \end{vmatrix}\]
• Compute the determinant to find the vector \(\mathbf{c} = (c_1, c_2, c_3)\) where each component is derived from subtracting products of corresponding components.
• The result is a vector perpendicular to both \(\mathbf{a}\) and \(\mathbf{b}\).

Understanding the cross product is essential when dealing with problems that involve torque, rotational motion, and finding areas in three-dimensional space.

Scalar Triple Product

The scalar triple product is a scalar value that involves three vectors. It is used to determine the volume of the parallelepiped formed by the three vectors. If you have vectors \(\mathbf{a}, \mathbf{b},\) and \(\mathbf{c}\), the scalar triple product is calculated as \(\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})\).
This can be understood as taking the dot product of vector \(\mathbf{a}\) with the cross product of vectors \(\mathbf{b}\) and \(\mathbf{c}\).

• First, compute the cross product \(\mathbf{b} \times \mathbf{c}\), resulting in a vector \(\mathbf{d}\).
• Then, use the dot product \(\mathbf{a} \cdot \mathbf{d}\) to get a single scalar value.
• This scalar represents the signed volume of the parallelepiped, and its magnitude gives the actual volume.

This operation is very important in physics and engineering for volume calculations and to determine the orientation of vectors.

Determinant

The concept of a determinant comes from linear algebra and plays a crucial role in vector calculus. It is a scalar value that can be computed from the elements of a square matrix. The determinant provides vital information about the matrix, such as whether it is invertible, and is used in the calculation of the cross product.
For a 3x3 matrix, the determinant is calculated as:

• \[\begin{vmatrix} a & b & c \ d & e & f \ g & h & i \end{vmatrix} = a \cdot (ei - fh) - b \cdot (di - fg) + c \cdot (dh - eg)\]
• This expansion shows how the values are sifted and subtracted to result in the determinant value.

The determinant of vectors aligned in a matrix form simply evaluates the strength and direction of the vectors' influence on each other. In physical terms, if you calculate the determinant as part of the cross product, you're determining a vector orthogonal to the original two vectors. Understanding determinants allows you to perform various operations in mathematics, physics, and engineering with insights into geometrical dimensions and dynamics.