Problem 19 - Thomas Calculus

Question

In Exercises \(19-22,\) verify that \((\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w}=(\mathbf{v} \times \mathbf{w}) \cdot \mathbf{u}=\) \((\mathbf{w} \times \mathbf{u}) \cdot \mathbf{v}\) and find the volume of the parallelepiped (box) determined by \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w} .\) $$ \begin{array}{ll}{2 \mathbf{i}} & {2 \mathbf{j}} & {2 \mathbf{k}}\end{array} $$

Step-by-Step Solution

Answer

The volume is 8. The identity is verified as all dot products are 8.

Step 1: Vector Definitions

Let's define the vectors \( \mathbf{u} = 2\mathbf{i}, \mathbf{v} = 2\mathbf{j}, \mathbf{w} = 2\mathbf{k} \). We need to verify the given identity and compute the volume of the parallelepiped formed by these vectors.

Step 2: Calculate \(\mathbf{u} \times \mathbf{v}\)

The cross product \( \mathbf{u} \times \mathbf{v}\) is calculated using the determinant:\[ \mathbf{u} \times \mathbf{v} = | \begin{bmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ 2 & 0 & 0 \ 0 & 2 & 0 \end{bmatrix} | = (0)\mathbf{i} - (0)\mathbf{j} + (4)\mathbf{k} = 4\mathbf{k} \]

Step 3: Dot with \(\mathbf{w}\)

Calculate \((\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w}\): \[ (4\mathbf{k}) \cdot (2\mathbf{k}) = 4 \cdot 2 = 8 \]

Step 4: Calculate \(\mathbf{v} \times \mathbf{w}\)

The cross product \( \mathbf{v} \times \mathbf{w} \) is: \[ | \begin{bmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ 0 & 2 & 0 \ 0 & 0 & 2 \end{bmatrix} | = (4)\mathbf{i} - (0)\mathbf{j} + (0)\mathbf{k} = 4\mathbf{i} \]

Step 5: Dot with \(\mathbf{u}\)

Calculate \((\mathbf{v} \times \mathbf{w}) \cdot \mathbf{u}\): \[ (4\mathbf{i}) \cdot (2\mathbf{i}) = 4 \cdot 2 = 8 \]

Step 6: Calculate \(\mathbf{w} \times \mathbf{u}\)

The cross product \( \mathbf{w} \times \mathbf{u} \) is:\[ | \begin{bmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ 0 & 0 & 2 \ 2 & 0 & 0 \end{bmatrix} | = (0)\mathbf{i} - (4)\mathbf{j} + (0)\mathbf{k} = -4\mathbf{j} \]

Step 7: Dot with \(\mathbf{v}\)

Calculate \((\mathbf{w} \times \mathbf{u}) \cdot \mathbf{v}\): \[ (-4\mathbf{j}) \cdot (2\mathbf{j}) = -4 \cdot 2 = 8 \]

Step 8: Verify Identity

The results for \((\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w} = 8\), \((\mathbf{v} \times \mathbf{w}) \cdot \mathbf{u} = 8\), and \((\mathbf{w} \times \mathbf{u}) \cdot \mathbf{v} = 8\) confirm the identity is true as all are equal.

Step 9: Volume of Parallelepiped

The volume of the parallelepiped is given by the scalar triple product: \[ \text{Volume} = \mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) = 8 \] This matches our previous computations.

Key Concepts

Cross Product

The cross product is a fundamental operation in vector algebra. It involves taking two vectors in three-dimensional space and producing another vector that is orthogonal to both. This orthogonality property is useful in many applications, such as calculating areas and volumes.To find the cross product of two vectors \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} + a_3 \mathbf{k} \) and \( \mathbf{b} = b_1 \mathbf{i} + b_2 \mathbf{j} + b_3 \mathbf{k} \), we use the determinant:

• \( \mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \end{vmatrix} \)
• This results in \( (a_2b_3 - a_3b_2) \mathbf{i} - (a_1b_3 - a_3b_1) \mathbf{j} + (a_1b_2 - a_2b_1) \mathbf{k} \).

The cross product is also called the vector product. In the exercise, we calculate \( \mathbf{u} \times \mathbf{v} \), \( \mathbf{v} \times \mathbf{w} \), and \( \mathbf{w} \times \mathbf{u} \) to verify an important vector identity.

Dot Product

The dot product, or scalar product, is another key vector operation. Unlike the cross product, it results in a scalar quantity. It measures how much one vector points in the direction of another, analogous to calculating the projection of one vector onto another. Given vectors \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} + a_3 \mathbf{k} \) and \( \mathbf{b} = b_1 \mathbf{i} + b_2 \mathbf{j} + b_3 \mathbf{k} \), the dot product is calculated as:

• \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \)

In our example, when we took the dot product after calculating the cross products, such as \( (\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w} \), the resultant scalar value helped us verify the given identity. This operation tells us how much of vector \( \mathbf{w} \) lies in the direction of \( \mathbf{u} \times \mathbf{v} \), a crucial aspect in understanding vector interactions.

Parallelepiped Volume

The volume of a parallelepiped, a six-faced figure where each face is a parallelogram, can be determined using vectors. Specifically, the volume is given by the magnitude of the scalar triple product of the three vectors defining the parallelepiped.If you have vectors \( \mathbf{u}, \mathbf{v}, \) and \( \mathbf{w} \), the volume \( V \) can be computed as:

• \( V = |\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})| \)

This expression essentially captures how the vectors span the three-dimensional space to enclose the volume. By taking the dot product of one vector with the perpendicular area formed by the cross product of the other two, you naturally determine the volume. In our exercise, where each vector is a simple scalar multiple of unit vectors, the volume equates to 8, matching our calculation made earlier.

Scalar Triple Product

The scalar triple product combines the cross product and dot product to yield a scalar value. It's used extensively in calculating volumes and verifying vector identities.Given three vectors \( \mathbf{u}, \mathbf{v}, \) and \( \mathbf{w} \), the scalar triple product is given by:

• \( \mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) \)
• Alternatively, it can be expressed as the determinant of a matrix formed by the three vectors:\( \begin{vmatrix} u_1 & u_2 & u_3 \ v_1 & v_2 & v_3 \ w_1 & w_2 & w_3 \end{vmatrix} \)

The scalar triple product zero indicates coplanar vectors, meaning they lie in the same geometric plane. Otherwise, the absolute value gives the volume of the parallelepiped spanned by the vectors. In our exercise, this was calculated as 8, effectively confirming the vectors form a solid in space with that volume.