Problem 34 - Precalculus Mathematics for Calculus

Question

Three vectors \(\mathbf{a}\), \(\mathbf{b}\), and \(\mathbf{c}\) are given. \(\mathbf{(a)}\) Find their scalar triple product \(\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c}) .\) \(\mathbf{(b)}\) Are the vectors coplanar? If not, find the volume of the parallelepiped that they determine. $$\mathbf{a}=2 \mathbf{i}-2 \mathbf{j}-3 \mathbf{k}, \quad \mathbf{b}=3 \mathbf{i}-\mathbf{j}-\mathbf{k}, \quad \mathbf{c}=6 \mathbf{i}$$

Step-by-Step Solution

Answer

The scalar triple product is -30; vectors are not coplanar, and the volume is 30.

Step 1: Find the cross product \( \mathbf{b} \times \mathbf{c} \)

The cross product \( \mathbf{b} \times \mathbf{c} \) is computed using the determinant formula where \( \mathbf{b} = 3\mathbf{i} - \mathbf{j} - \mathbf{k} \) and \( \mathbf{c} = 6\mathbf{i} \). \[\mathbf{b} \times \mathbf{c} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ 3 & -1 & -1 \ 6 & 0 & 0 \end{vmatrix}\]Expanding the determinant:\[\mathbf{b} \times \mathbf{c} = \mathbf{i}(0 + 0) - \mathbf{j}(-3 \times 0 - (-1) \times 6) + \mathbf{k}(3 \times 0 - (-1) \times 6)\]\[\mathbf{b} \times \mathbf{c} = 0\mathbf{i} + 6\mathbf{j} + 6\mathbf{k} = 6\mathbf{j} + 6\mathbf{k}\]

Step 2: Calculate the scalar triple product \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \)

Use the vector \( \mathbf{a} = 2\mathbf{i} - 2\mathbf{j} - 3\mathbf{k} \) and the result from the previous step \( \mathbf{b} \times \mathbf{c} = 6\mathbf{j} + 6\mathbf{k} \). The scalar triple product is:\[\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = (2\mathbf{i} - 2\mathbf{j} - 3\mathbf{k}) \cdot (6\mathbf{j} + 6\mathbf{k})\]Calculate the dot product:\[\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = 2 \cdot 0 + (-2) \cdot 6 + (-3) \cdot 6\]\[\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = 0 - 12 - 18 = -30\]

Step 3: Determine if vectors are coplanar

Vectors are coplanar if their scalar triple product is zero. Since we found:\[\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = -30\]The scalar triple product is not zero, so the vectors are not coplanar.

Step 4: Find the volume of the parallelepiped

The volume of the parallelepiped formed by the vectors \( \mathbf{a} \), \( \mathbf{b} \), and \( \mathbf{c} \) is equal to the absolute value of the scalar triple product:\[\text{Volume} = |\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})| = |-30| = 30\]

Key Concepts

Vector Cross Product

In vector mathematics, the cross product of two vectors results in another vector that is perpendicular to the plane formed by the original two vectors. This is essential when calculating areas and volumes in three-dimensional space. Mathematically, for vectors \( \mathbf{b} = 3\mathbf{i} - \mathbf{j} - \mathbf{k} \) and \( \mathbf{c} = 6\mathbf{i} \), the cross product \( \mathbf{b} \times \mathbf{c} \) makes use of the determinant of a matrix formed by the unit vectors \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) and the coefficients of the vectors from their components.

• Visualize the physical scenario: If you imagine flat hands aligned with \( \mathbf{b} \) and \( \mathbf{c} \), the thumb that sticks out perpendicularly represents \( \mathbf{b} \times \mathbf{c} \).

For this exercise, using the determinant method, we computed \( \mathbf{b} \times \mathbf{c} = 6\mathbf{j} + 6\mathbf{k} \). This resulting vector completes the necessary calculations for volume determination in subsequent steps.

Dot Product

The dot product, also known as the scalar product, is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number. The dot product of vectors \( \mathbf{a} \) and \( \mathbf{b} \) is found by multiplying corresponding coordinates and then summing those products.

• Use case: It quantifies how much one vector extends in the direction of another, which is useful when determining if vectors span the same region.

For the scalar triple product \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \), you first need to compute \( \mathbf{b} \times \mathbf{c} \), and then dot this result with vector \( \mathbf{a} \). Applying this to the given vectors, it was calculated as \(-30\), which is crucial for the volume finding in vector analysis.

Coplanar Vectors

Vectors are considered coplanar if they lie within the same three-dimensional space or plane. This often implies that one of the vectors can be expressed as a linear combination of the other two. A quick way to check for coplanarity is to compute their scalar triple product.

• If the scalar triple product is zero, the vectors are coplanar.
• In cases when the scalar triple product is non-zero, like here, the vectors are not coplanar.

Since our computed scalar triple product was \(-30\), it indicates these vectors do not lie on the same plane and they define a three-dimensional space, progressing towards finding the volume of a shape as result.

Parallelepiped Volume

The volume of a parallelepiped, which is a three-dimensional figure formed by three vectors, is calculated by taking the absolute value of the scalar triple product. Basically, the scalar triple product gives you the signed volume, where the sign indicates direction but not size. The magnitude shows the actual volume.

• Formula derived: \( \text{Volume} = |\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})| \).
• Real-world analogy: Think of this as measuring the capacity of a tilted box or 3D parallelogram.

From the exercise, we concluded that the absolute volume is \(30\). This is because the calculated scalar triple product was \(-30\), and thus the magnitude, or absolute value, results in the positive volume of the parallelepiped, showing us the three-dimensional space the vectors enclose.