Problem 5 - An Introduction to the Mechanics of Solids in SI Units

Question

Let \(\mathbf{F}\) be an arbitrary vector through the origin and let \(\mathbf{a}, \mathbf{b}, \mathbf{c}\) be three arbitrary noncoplanar unit vectors passing through the origin. It is desired to decompose \(\mathbf{F}\) into vectors parallel to \(\mathbf{a}, \mathbf{b}\), and \(\mathbf{c} ;\) that is, it is desired to find magnitudes \(L_{a}, L_{b}\), and \(L_{c}\) such that $$ \mathbf{F}=L_{a} \mathbf{a}+L_{b} \mathbf{b}+L_{c} \mathbf{c} $$ Sketch this and show that it involves a parallelepiped whose edges are parallel to \(\mathbf{a}, \mathbf{b}\), and \(\mathbf{c}\) with \(\mathbf{F}\) as the diagonal. Use the properties of the scalar triple product to show that $$ L_{a}=\frac{\mathbf{F} \times \mathbf{b} \cdot \mathbf{c}}{\mathbf{a} \times \mathbf{b} \cdot \mathbf{c}} $$ with similar expressions for \(L_{b}\) and \(L_{c^{\circ}}\)

Step-by-Step Solution

Answer

The magnitudes \(L_{a}, L_{b}\), and \(L_{c}\) that decompose the vector \(\mathbf{F}\) into vectors parallel to \(\mathbf{a}, \mathbf{b}, \mathbf{c}\) are given by \(L_{a}=\frac{\mathbf{F} \times \mathbf{b} . \mathbf{c}}{\mathbf{a} . \mathbf{b} . \mathbf{c}}\), \(L_{b}=\frac{\mathbf{F} \times \mathbf{c} . \mathbf{a}}{\mathbf{a} . \mathbf{b} . \mathbf{c}}\), and \(L_{c}=\frac{\mathbf{F} \times \mathbf{a} . \mathbf{b}}{\mathbf{a} . \mathbf{b} . \mathbf{c}}\), respectively.

Step 1: Representation

Represent the vector \(\mathbf{F}\) as a result of vector addition i.e. \(\mathbf{F}=L_{a} \mathbf{a}+L_{b} \mathbf{b}+L_{c} \mathbf{c}\). This can be visualized as constructing a parallelepiped with \(\mathbf{a}, \mathbf{b}\), and \(\mathbf{c}\) as the edges and \(\mathbf{F}\) as the diagonal.

Step 2: Scalar Triple Product

Use the properties of the scalar triple product. Remember, the scalar triple product of three vectors \(\mathbf{a}, \mathbf{b}\) and \(\mathbf{c}\) is defined as \(\mathbf{a} . (\mathbf{b} \times \mathbf{c})\), which equal to the volume of the parallelepiped built on them.

Step 3: Decompose the Vector F

Decompose the vector \(\mathbf{F}\) into components. First, find the component of \(\mathbf{F}\) parallel to \(\mathbf{a}\), denoted as \(L_{a}\). We can get it as \(L_{a}=\frac{\mathbf{F} \times \mathbf{b} . \mathbf{c}}{\mathbf{a} \times \mathbf{b} . \mathbf{c}}\).

Step 4: Find Other Components

By symmetry, the components of \(\mathbf{F}\) parallel to \(\mathbf{b}\) and \(\mathbf{c}\), denoted as \(L_{b}\) and \(L_{c}\) respectively, can be found as \(L_{b}=\frac{\mathbf{F} \times \mathbf{c} . \mathbf{a}}{\mathbf{a} . \mathbf{b} \times \mathbf{c}}\) and \(L_{c}=\frac{\mathbf{F} \times \mathbf{a} . \mathbf{b}}{\mathbf{a} . \mathbf{b} \times \mathbf{c}}\).

Key Concepts

Scalar Triple Product

The scalar triple product is a significant mathematical tool used in vector analysis. It involves three vectors and is defined as \[ \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \]This operation results in a scalar value, not another vector. The scalar triple product is particularly useful for finding the volume of a parallelepiped constructed by the three vectors. In simpler terms, if you imagine building a box using vectors \(\mathbf{a}, \mathbf{b},\) and \(\mathbf{c}\) as edges, the triple product will tell you the amount of space contained within the box.

• If the scalar triple product is zero, it indicates that the vectors lie in the same plane, hence forming a degenerate parallelepiped with zero volume.
• The scalar triple product is also used to determine if the vectors \(\mathbf{a}, \mathbf{b},\) and \(\mathbf{c}\) are coplanar.

In the given exercise, we use this product to derive expressions for the components of vector \(\mathbf{F}\), which is decomposed along the directions of \(\mathbf{a}, \mathbf{b},\) and \(\mathbf{c}\).

Parallelepiped

Think of a parallelepiped as an elongated box or a three-dimensional figure formed by six parallelograms. It is an important visualization in understanding the decomposition of vectors. In this exercise, the parallelepiped is formed by three unit vectors \(\mathbf{a}, \mathbf{b},\) and \(\mathbf{c}\), serving as its edges. The vector \(\mathbf{F}\) then acts as the diagonal across this shape.
This geometric concept is tightly linked with the scalar triple product, since the volume of the parallelepiped is precisely the value of the scalar triple product \(\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})\). Here:

• The edges \(\mathbf{a}, \mathbf{b},\) and \(\mathbf{c}\) depict the span of the structure.
• The diagonal \(\mathbf{F}\) represents the combination of these vectors in space.

Understanding the parallelepiped helps in visualizing how the scalar triple product operates within a given set of vectors, providing clarity to how the magnitudes \(L_a, L_b,\) and \(L_c\) are derived.

Vector Components

Vector components are the pieces of a vector that point in the direction of the unit vectors of a coordinate system. More simply, any vector can be expressed as a sum of other vectors each aligned with a coordinate axis, such as \(\mathbf{i}, \mathbf{j},\) and \(\mathbf{k}\).
In the exercise, decomposing the vector \(\mathbf{F}\) involves breaking it down into parts that are parallel to \(\mathbf{a}, \mathbf{b},\) and \(\mathbf{c}\). These parts are referred to as \(L_a \mathbf{a}, L_b \mathbf{b},\) and \(L_c \mathbf{c}\), respectively.

• The magnitude of the component \(L_a\) is calculated using the formula \(L_a = \frac{\mathbf{F} \times \mathbf{b} \cdot \mathbf{c}}{\mathbf{a} \times \mathbf{b} \cdot \mathbf{c}}\).
• Similarly, components \(L_b\) and \(L_c\) can be found by swapping the positions of the unit vectors \(\mathbf{a}, \mathbf{b},\) and \(\mathbf{c}\) in the scalar triple product relationships.

This decomposition process allows us to perceive the vector \(\mathbf{F}\) in terms of its direction and magnitude relative to each of the unit vectors.

Unit Vectors

Unit vectors are vectors with a magnitude of one. They are crucial in decomposing vectors because they provide the directionality in vector operations without affecting the magnitude.
In this context, \(\mathbf{a}, \mathbf{b},\) and \(\mathbf{c}\) are unit vectors that serve as reference directions in space.

• They are often denoted with a hat symbol (\(\mathbf{\hat{a}}, \mathbf{\hat{b}}, \mathbf{\hat{c}}\)) to differentiate them from other vectors.
• Unit vectors are a building block for expressing any vector, as every vector can be decomposed into components along these directions.
• They help form standard bases in vector spaces, such that any point or object can be described simply by its coordinates in relation to these directions.

Understanding unit vectors solidifies the foundation for vector decomposition, as they simplify calculations and provide a clear sense of direction when working with complex vectors.