Problem 34 - Essential Calculus Early Transcendentals

Question

Find the volume of the parallelepiped determined by the vectors a, \(\mathbf{b},\) and \(\mathbf{c} .\) $$\mathbf{a}=\mathbf{i}+\mathbf{j}, \quad \mathbf{b}=\mathbf{j}+\mathbf{k}, \quad \mathbf{c}=\mathbf{i}+\mathbf{j}+\mathbf{k}$$

Step-by-Step Solution

Answer

The volume of the parallelepiped is 1.

Step 1: Recall the Volume Formula

The volume of a parallelepiped determined by vectors \( \mathbf{a}, \mathbf{b}, \text{ and } \mathbf{c} \) is given by the scalar triple product \( | \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) | \). We need to compute the cross product \( \mathbf{b} \times \mathbf{c} \) first and then find the dot product with \( \mathbf{a} \).

Step 2: Find the Cross Product \( \mathbf{b} \times \mathbf{c} \)

Given \( \mathbf{b} = \mathbf{j} + \mathbf{k} \) and \( \mathbf{c} = \mathbf{i} + \mathbf{j} + \mathbf{k} \), calculate the cross product:\[\mathbf{b} \times \mathbf{c} = \begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \0 & 1 & 1 \1 & 1 & 1\end{vmatrix} = \mathbf{i}(1 - 1) - \mathbf{j}(0 - 1) + \mathbf{k}(0 - 1) = \mathbf{j} - \mathbf{k}.\]

Step 3: Compute the Dot Product \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \)

Now calculate the dot product between \( \mathbf{a} = \mathbf{i} + \mathbf{j} \) and \( \mathbf{b} \times \mathbf{c} = \mathbf{j} - \mathbf{k} \):\[\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = (\mathbf{i} + \mathbf{j}) \cdot (\mathbf{j} - \mathbf{k}) = 1 \times 0 + 1 \times 1 = 1.\]

Step 4: Calculate the Volume

The volume of the parallelepiped is the absolute value of the scalar triple product: \(|\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})| = |1| = 1.\)

Key Concepts

Scalar Triple Product

The scalar triple product is a fundamental concept for calculating the volume of a parallelepiped defined by three vectors \( \mathbf{a}, \mathbf{b}, \) and \( \mathbf{c} \). The formula used is \( | \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})| \). This expression includes two operations: the cross product and the dot product.
The absolute value of this product provides the volume. It effectively measures how much space the parallelepiped occupies.

• The expression \( (\mathbf{b} \times \mathbf{c}) \) finds a vector perpendicular to both \( \mathbf{b} \) and \( \mathbf{c} \).
• The dot product then finds how much the vector \( \mathbf{a} \) aligns with this perpendicular vector.
• The absolute value ensures the volume is non-negative.

Calculating the scalar triple product helps confirm how the vectors contribute to the shape in three-dimensional space. This calculation is crucial because if the volume is zero, it indicates the vectors are coplanar.

Cross Product

The cross product is an operation on two vectors in three-dimensional space, resulting in another vector. This operation is essential when working with parallelepipeds as it provides a vector perpendicular to both original vectors. The cross product of two vectors \( \mathbf{b} \) and \( \mathbf{c} \) is denoted as \( \mathbf{b} \times \mathbf{c} \).
For vectors \( \mathbf{b} = \mathbf{j} + \mathbf{k} \) and \( \mathbf{c} = \mathbf{i} + \mathbf{j} + \mathbf{k} \), we find the cross product by setting up a determinant matrix:
\[\mathbf{b} \times \mathbf{c} = \begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \0 & 1 & 1 \1 & 1 & 1\end{vmatrix} = \mathbf{j} - \mathbf{k}.\]
This result—a new vector \( \mathbf{j} - \mathbf{k} \)—is normal to both \( \mathbf{b} \) and \( \mathbf{c} \), making it an essential component in the scalar triple product calculation. Remember, the cross product is always perpendicular to the plane formed by the original vectors, and its magnitude represents the area of the parallelogram formed by them.

Dot Product

The dot product provides a way to multiply two vectors, resulting in a scalar. This result represents the magnitude of projection of one vector onto another. It's crucial for confirming if vectors align and consequently for calculating volume when combined with a cross product.
When computing the volume of a parallelepiped, the dot product involves a vector \( \mathbf{a} \) and the vector result of a cross product \( \mathbf{b} \times \mathbf{c} \). In our example, \( \mathbf{a} = \mathbf{i} + \mathbf{j} \) and \( \mathbf{b} \times \mathbf{c} = \mathbf{j} - \mathbf{k} \). The calculation proceeds as follows:
\[\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = (\mathbf{i} + \mathbf{j}) \cdot (\mathbf{j} - \mathbf{k}) = 0 \cdot 0 + 1 \cdot 1 - 0 = 1.\]
The outcome is a single number, in this case, 1, which can directly influence the calculated volume. The dot product helps ascertain the alignment of vectors and their resulting combination determined by the cross product. In essence, it guarantees the precision of spatial computations by quantifying vector directionality.