Question

Given three vectors \(\boldsymbol{a}, \boldsymbol{b}\) and \(\boldsymbol{c}\), their triple scalar product is defined to be \((\boldsymbol{a} \times \boldsymbol{b}) \cdot \boldsymbol{c}\). It can be shown that the modulus of this is the volume of the parallelepiped formed by the three vectors. Find the volume of the parallelepiped formed by the three vectors \(\boldsymbol{a}=3 \boldsymbol{i}+\boldsymbol{j}-2 \boldsymbol{k}, \boldsymbol{b}=\boldsymbol{i}+2 \boldsymbol{j}-2 \boldsymbol{k}\) and \(\boldsymbol{c}=2 \boldsymbol{i}+5 \boldsymbol{j}+\boldsymbol{k}\)

Short answer

Answer: The volume of the parallelepiped is 24 cubic units.

Step-by-step solution

Compute the cross product of \(\boldsymbol{a}\) and \(\boldsymbol{b}\)

First, we need to calculate the cross product \(\boldsymbol{a} \times \boldsymbol{b}\). The cross product of two vectors can be found using the determinant of a 3x3 matrix: $$ \boldsymbol{a} \times \boldsymbol{b} = \begin{vmatrix} \boldsymbol{i} & \boldsymbol{j} & \boldsymbol{k} \\ 3 & 1 & -2 \\ 1 & 2 & -2 \end{vmatrix} $$ Compute the determinant using the standard cofactor expansion method. The result is: $$ \boldsymbol{a} \times \boldsymbol{b} = (4-4)\boldsymbol{i} - (-6+2)\boldsymbol{j} + (6-2)\boldsymbol{k} = 0\boldsymbol{i}+4\boldsymbol{j}+4\boldsymbol{k} $$

Compute the dot product of \((\boldsymbol{a} \times \boldsymbol{b})\) and \(\boldsymbol{c}\)

Next, we need to compute the dot product of the cross product we found in the previous step and \(\boldsymbol{c}\). The dot product is calculated as \(\sum_{i=1}^3 (A_i*B_i)\), where \(A_i\) and \(B_i\) are the corresponding components of the vectors. So, we have: $$ (\boldsymbol{a} \times \boldsymbol{b}) \cdot \boldsymbol{c} = 0(2) + 4(5) + 4(1) = 20 + 4 = 24 $$

Find the modulus of the triple scalar product

The volume of the parallelepiped formed by the three vectors is the modulus of the triple scalar product, i.e., \(|(\boldsymbol{a} \times \boldsymbol{b}) \cdot \boldsymbol{c}|\). In this case, the triple scalar product is already a positive value, so the modulus remains unchanged: $$ |24| = 24 $$

State the final answer

The volume of the parallelepiped formed by the vectors \(\boldsymbol{a}, \boldsymbol{b},\) and \(\boldsymbol{c}\) is 24 cubic units.

Key concepts

Triple Scalar Product

The triple scalar product is a fascinating concept in vector calculus. It involves three vectors, say \(\boldsymbol{a}, \boldsymbol{b}, \text{ and } \boldsymbol{c}\), and is calculated using the formula \((\boldsymbol{a} \times \boldsymbol{b}) \cdot \boldsymbol{c}\). This operation starts with a cross product followed by a dot product.

By computing the cross product of two vectors from the set, we create a new vector that is orthogonal to the original two vectors. Then, taking the dot product of this new vector with the third vector gives a scalar result. This scalar value represents a specific geometric measurement, which brings us to our next topic.

Cross Product

The cross product is an operation on two vectors in three-dimensional space, resulting in another vector. This new vector is perpendicular to the plane defined by the original two vectors.

To compute the cross product, arrange the components of the two vectors and unit vectors \(\boldsymbol{i, j, k}\) in a 3x3 matrix and find the determinant. For vectors \(\boldsymbol{a} = 3\boldsymbol{i} + \boldsymbol{j} - 2\boldsymbol{k}\) and \(\boldsymbol{b} = \boldsymbol{i} + 2\boldsymbol{j} - 2\boldsymbol{k}\), the cross product is:

\[\boldsymbol{a} \times \boldsymbol{b} = \begin{vmatrix} \boldsymbol{i} & \boldsymbol{j} & \boldsymbol{k} \ 3 & 1 & -2 \ 1 & 2 & -2 \end{vmatrix}\]

This simplifies to \(0\boldsymbol{i} + 4\boldsymbol{j} + 4\boldsymbol{k}\). The result is a vector that is orthogonal to both \(\boldsymbol{a}\) and \(\boldsymbol{b}\).

Dot Product

The dot product, unlike the cross product, results in a scalar value. It measures the cosine of the angle between two vectors multiplied by their magnitudes.

For our problem, we take the dot product of the cross product vector \((4\boldsymbol{j} + 4\boldsymbol{k})\) and the third vector \(\boldsymbol{c} = 2\boldsymbol{i} + 5\boldsymbol{j} + \boldsymbol{k}\). Calculating the dot product involves multiplying corresponding components and summing these products:

\[(0*2) + (4*5) + (4*1) = 0 + 20 + 4 = 24\]

The result, 24, signifies the aligned magnitude of these vectors.

Parallelepiped Volume

The concept of the parallelepiped volume is closely linked to the triple scalar product. A parallelepiped is a three-dimensional figure with planar faces, sometimes imagined as a skewed box.

The volume of a parallelepiped formed by vectors \(\boldsymbol{a}, \boldsymbol{b},\) and \(\boldsymbol{c}\) is given by the absolute value of their triple scalar product. This underlying principle connects vector operations to a tangible geometric form. In our example, we calculated this value to be 24 cubic units, indicating a sizeable space occupied by the vectors in a 3D plane.

Understanding this relation not only encompasses a fundamental aspect of vector calculus but also grounds abstract mathematical operations in real-world shapes.