Problem 51 - University Calculus: Early Transcendentals

Question

Using the methods of Section \(6.1,\) where volume is computed by integrating cross-sectional area, it can be shown that the volume of a tetrahedron formed by three vectors is equal to \(\frac{1}{6}\) the volume of the parallelipiped formed by the three vectors. Find the volumes of the tetrahedra whose vertices are given. $$A(0,0,0), \quad B(2,0,0), \quad C(0,3,0), \quad D(0,0,4)$$

Step-by-Step Solution

Answer

The volume of the tetrahedron is 4.

Step 1: Identify the vectors from the vertices

The vectors forming the sides of the tetrahedron can be derived by subtracting coordinates of the given points. These vectors are:- \( \vec{AB} = B - A = (2 - 0, 0 - 0, 0 - 0) = (2, 0, 0) \)- \( \vec{AC} = C - A = (0 - 0, 3 - 0, 0 - 0) = (0, 3, 0) \)- \( \vec{AD} = D - A = (0 - 0, 0 - 0, 4 - 0) = (0, 0, 4) \)

Step 2: Determine the parallelepiped volume

The volume of the parallelepiped formed by these vectors is given by the scalar triple product \( \vec{AB} \cdot (\vec{AC} \times \vec{AD}) \). To compute this, find the cross product and then dot product:Cross Product \( \vec{AC} \times \vec{AD} \):\[\begin{vmatrix}\hat{i} & \hat{j} & \hat{k} \0 & 3 & 0 \0 & 0 & 4\end{vmatrix} = (12\hat{i} - 0\hat{j} + 0\hat{k}) = (12, 0, 0)\]Dot Product \( \vec{AB} \cdot (\vec{AC} \times \vec{AD}) \):\[(2, 0, 0) \cdot (12, 0, 0) = 24\]Thus, the volume of the parallelepiped is 24.

Step 3: Compute the tetrahedron volume

The formula for the volume of a tetrahedron formed by vectors is \( \frac{1}{6} \) of the volume of the parallelepiped. With a parallelepiped volume of 24, the tetrahedron volume is:\[Volume = \frac{1}{6} \times 24 = 4\]

Key Concepts

Vector Operations

Vector operations involve basic arithmetic calculations with vectors, which are entities possessing both magnitude and direction. In geometry and physics, vectors are represented as arrows in space, having specific components along coordinate axes. Basic vector operations include addition, subtraction, and scalar multiplication.

• Addition: Vectors are added component-wise. For example, if you have vectors \( \vec{a} = (a_1, a_2, a_3) \) and \( \vec{b} = (b_1, b_2, b_3) \), their sum is \( \vec{a} + \vec{b} = (a_1 + b_1, a_2 + b_2, a_3 + b_3) \).
• Subtraction: Similar to addition, vector subtraction is done component-wise: \( \vec{a} - \vec{b} = (a_1 - b_1, a_2 - b_2, a_3 - b_3) \).
• Scalar Multiplication: This operation scales a vector by multiplying each component by a scalar. If \( c \) is a scalar, then \( c\vec{a} = (ca_1, ca_2, ca_3) \).

Understanding these operations is crucial, as they are the foundation for more complex vector calculations, such as the cross product and the scalar triple product.

Cross Product

The cross product is a unique operation possible only between two three-dimensional vectors. It results in another vector that is perpendicular to the plane containing the original vectors. The magnitude of the cross product vector is equal to the area of the parallelogram spanned by the original vectors. Finding this product follows a specific method involving a determinant.

To compute the cross product \( \vec{a} \times \vec{b} \), construct a 3x3 matrix using the unit vectors \( \hat{i}, \hat{j}, \hat{k} \) in the top row, followed by the components of \( \vec{a} \) and \( \vec{b} \): \[\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \end{vmatrix} \] This results in the vector \((a_2b_3 - a_3b_2)\hat{i} - (a_1b_3 - a_3b_1)\hat{j} + (a_1b_2 - a_2b_1)\hat{k}\).

• Ensure the vectors are correctly represented in R3, highlighting the axes they manipulate.
• The resulting vector's direction is governed by the right-hand rule.

The cross product is essential for deriving the scalar triple product needed for volume calculations in geometry.

Scalar Triple Product

The scalar triple product is a mathematical operation involving three vectors resulting in a scalar value. This scalar is essential for computing the volume of a parallelepiped defined by three vectors. Conceptually, it combines both the dot product and cross product in a specific sequence.

The operation is expressed as \( \vec{a} \cdot (\vec{b} \times \vec{c}) \). Here's how it works:

• First, compute the cross product \( \vec{b} \times \vec{c} \), resulting in a vector perpendicular to both \( \vec{b} \) and \( \vec{c} \).
• Next, the dot product of \( \vec{a} \) with this resulting vector gives a scalar value, representing the volume of the parallelepiped.

For the tetrahedron, this scalar is divided by 6, as a tetrahedron is a sixth of the parallelepiped. It is crucial to follow the correct order since changing the sequence of the vectors could alter the volume sign, signifying a reversed orientation. This operation underscores the elegance of vector mathematics in simplifying 3D geometry problems, connecting algebraic computations to spatial concepts.