Problem 1 - A First Course in Continuum Mechanics

Question

Given the vectors \(\boldsymbol{a}=1 \boldsymbol{i}+2 \boldsymbol{j}+3 \boldsymbol{k}, \boldsymbol{b}=1 \boldsymbol{i}+3 \boldsymbol{j}-2 \boldsymbol{k}\) and \(c=-2 i-1 j+0 k\), calculate: (a) \(a \cdot b\) (b) \(a \times b\) (c) \(a \cdot b \times c\) (d) \(\boldsymbol{a} \times(\boldsymbol{b} \times \boldsymbol{c})\), (e) \((\boldsymbol{a} \times \boldsymbol{b}) \times \boldsymbol{c}\).

Step-by-Step Solution

Answer

\( (a)~1, (b)~\boldsymbol{i} + 5\boldsymbol{j} + \boldsymbol{k}, (c)~0, (d)~\boldsymbol{i} + 5\boldsymbol{j} + \boldsymbol{k}, (e)~\boldsymbol{i} + 5\boldsymbol{j} + \boldsymbol{k}. \)

Step 1 - Calculate Dot Product (a ⋅ b)

The dot product of two vectors is found by multiplying their corresponding components and adding the products together. For vectors \(\boldsymbol{a}=1\boldsymbol{i}+2\boldsymbol{j}+3\boldsymbol{k}\) and \(\boldsymbol{b}=1\boldsymbol{i}+3\boldsymbol{j}-2\boldsymbol{k}\), the dot product \(a \cdot b\) is calculated as:\[a \cdot b = (1)(1) + (2)(3) + (3)(-2)\].

Step 2 - Calculate Cross Product (a × b)

To find the cross product of two vectors, you can use the determinant of a 3x3 matrix whose first row is the unit vectors \(\boldsymbol{i}, \boldsymbol{j}, \boldsymbol{k}\), the second row is the components of vector \(\boldsymbol{a}\), and the third row is the components of vector \(\boldsymbol{b}\). Calculate this determinant to find the cross product \(a × b\).

Step 3 - Calculate Triple Product (a ⋅ b × c)

The scalar triple product is given by \(\boldsymbol{a} \cdot (\boldsymbol{b} \times \boldsymbol{c})\). First, calculate the vector product \(\boldsymbol{b} \times \boldsymbol{c}\), then take the dot product of that result with vector \(\boldsymbol{a}\).

Step 4 - Calculate Vector Triple Product (a × (b × c))

The vector triple product \(\boldsymbol{a} \times (\boldsymbol{b} \times \boldsymbol{c})\) is calculated by first finding the cross product \(\boldsymbol{b} \times \boldsymbol{c}\), then taking the cross product of vector \(\boldsymbol{a}\) with this result.

Step 5 - Calculate Double Cross Product ((a × b) × c)

For the double cross product \((\boldsymbol{a} \times \boldsymbol{b}) \times \boldsymbol{c}\), you first calculate the cross product \(\boldsymbol{a} \times \boldsymbol{b}\), and then take the cross product of this result with vector \(\boldsymbol{c}\).

Key Concepts

Dot Product

The dot product, also known as scalar product, is a fundamental operation in vector algebra, especially in the field of continuum mechanics. It's a way of multiplying two vectors that results in a scalar, rather than another vector. This operation is crucial for understanding projections, angles, and the length of vectors.

Mathematically, the dot product of two vectors \( \boldsymbol{a} \text{ and } \boldsymbol{b} \) is calculated by summing the products of their corresponding components. For instance, if \( \boldsymbol{a} = a_1 \boldsymbol{i} + a_2 \boldsymbol{j} + a_3 \boldsymbol{k} \) and \( \boldsymbol{b} = b_1 \boldsymbol{i} + b_2 \boldsymbol{j} + b_3 \boldsymbol{k} \) then the dot product is \(a \cdot b = a_1 b_1 + a_2 b_2 + a_3 b_3\).

This operation is commutative, meaning \(a \cdot b = b \cdot a\), and it's related to the cosine of the angle between the two vectors, so it can provide information about their directional relationship. In continuum mechanics, the dot product is used to find work done, project one vector onto another, and determine orthogonality between vectors.

Cross Product

The cross product is another essential vector operation in continuum mechanics, and unlike the dot product, it results in a vector, not a scalar. The cross product of two vectors \( \boldsymbol{a} \) and \( \boldsymbol{b} \) is a vector that is perpendicular to the plane containing \( \boldsymbol{a} \) and \( \boldsymbol{b} \). It has a magnitude equal to the area of the parallelogram formed by the two vectors.

The cross product, expressed as \( \boldsymbol{a} \times \boldsymbol{b} \) can be calculated using the determinant of a matrix, where the first row contains the unit vectors \( \boldsymbol{i}, \boldsymbol{j}, \boldsymbol{k} \), the second and third rows contain the components of vectors \( \boldsymbol{a} \) and \( \boldsymbol{b} \) respectively. This product is also direction-sensitive, following the right-hand rule, which means if you curl the fingers of your right hand from \( \boldsymbol{a} \) to \( \boldsymbol{b} \) the thumb points in the direction of the product \( \boldsymbol{a} \times \boldsymbol{b} \).

In practice, the cross product is used in continuum mechanics to determine torques, rotational effects, and is also relevant in describing the angular velocity of rotating bodies.

Scalar Triple Product

The scalar triple product is used when dealing with three vectors in continuum mechanics, and it's the result of performing a dot product and a cross product in sequence. It takes three vectors— let's call them \( \boldsymbol{a}, \boldsymbol{b}, \text{ and } \boldsymbol{c} \) — and results in a scalar value. The operation is written as \( \boldsymbol{a} \cdot (\boldsymbol{b} \times \boldsymbol{c}) \).

Geometrically, the magnitude of the scalar triple product is equal to the volume of the parallelepiped defined by the three vectors. Because of this property, the scalar triple product can be used to determine if the set of vectors are coplanar—if the result is zero, the vectors lie in the same plane. The orientation of this operation is critical, as it follows the right-hand rule similar to the cross product, and changing the order of the vectors can change the sign of the result.

In applications, the scalar triple product appears in fluid mechanics to calculate flow rate through a surface and in structural engineering to determine volumes and verify geometric configurations.

Vector Triple Product

The vector triple product involves three vectors and the operations of both the dot and cross product. This operation is not as commonly encountered as the previous ones but still bears importance in certain mechanics applications. The vector triple product is defined as \( \boldsymbol{a} \times (\boldsymbol{b} \times \boldsymbol{c}) \) and the result is a vector.

One interesting property of this operation is that it doesn't result in a vector that is necessarily perpendicular to all the original vectors. Instead, the outcome is a vector that lies in the plane spanned by \( \boldsymbol{b} \text{ and } \boldsymbol{c} \). The triple product can be used to derive vector components relative to a plane defined by other vectors and is also connected to the concept of the moment of a force.

Moreover, the vector triple product has an algebraic property that connects it back to the simpler operations of dot and cross product, known as the BAC-CAB rule: \( \boldsymbol{a} \times (\boldsymbol{b} \times \boldsymbol{c}) = \boldsymbol{b} (\boldsymbol{a} \cdot \boldsymbol{c}) - \boldsymbol{c} (\boldsymbol{a} \cdot \boldsymbol{b}) \). This rule simplifies the triple product calculation and often aids in analytical mechanics problems such as finding the equation of motion under certain force conditions.