Problem 31 - 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. $$\mathbf{a}=\langle 2,3,-2\rangle, \quad \mathbf{b}=\langle- 1,4,0\rangle, \quad \mathbf{c}=\langle 3,-1,3\rangle$$

Step-by-Step Solution

Answer

The scalar triple product is 59; vectors are not coplanar; volume is 59.

Step 1: Calculate Cross Product of b and c

To find the cross product \( \mathbf{b} \times \mathbf{c} \), use the determinant formula:\[\mathbf{b} \times \mathbf{c} = \begin{vmatrix}\hat{i} & \hat{j} & \hat{k} \-1 & 4 & 0 \3 & -1 & 3\end{vmatrix}\]This results in:\[(4 \cdot 3 - 0 \cdot -1)\hat{i} - (-1 \cdot 3 - 3 \cdot 0)\hat{j} + (-1 \cdot -1 - 4 \cdot 3)\hat{k}\]So, \( \mathbf{b} \times \mathbf{c} = \langle 12, 3, -13 \rangle \).

Step 2: Calculate Scalar Triple Product

Now, calculate the scalar triple product \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \).Given \( \mathbf{a} = \langle 2, 3, -2 \rangle \) and \( \mathbf{b} \times \mathbf{c} = \langle 12, 3, -13 \rangle \), the scalar triple product is:\[\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = 2 \cdot 12 + 3 \cdot 3 + (-2) \cdot (-13)\]Calculate each term: \( 24 + 9 + 26 = 59 \).Thus, the scalar triple product is \( 59 \).

Step 3: Determine if Vectors are Coplanar

Vectors are coplanar if their scalar triple product is equal to zero. Here, the scalar triple product is 59, which is not zero, indicating that \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) are not coplanar.

Step 4: Calculate Volume of Parallelepiped

The volume of the parallelepiped formed by \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) is the absolute value of the scalar triple product. Therefore, the volume is \( |59| = 59 \).

Key Concepts

Cross Product

The cross product is a mathematical operation used when dealing with three-dimensional vectors. It results in a vector that is perpendicular (orthogonal) to the original two vectors being multiplied.
To calculate the cross product of two vectors \(\mathbf{b}\) and \(\mathbf{c}\), you would arrange the vectors in a matrix alongside the unit vectors \( \hat{i}, \hat{j}, \hat{k} \). This is done using the determinant approach.

Here's how you calculate:

• Step 1: Form a 3x3 matrix with the first row as the unit vectors \(\hat{i}, \hat{j}, \hat{k}\).
• Step 2: Fill the second row with the components of vector \(\mathbf{b}\).
• Step 3: Fill the third row with the components of vector \(\mathbf{c}\).

Calculate the determinant to find the cross product, resulting in a new vector \(\mathbf{b} \times \mathbf{c}\). This vector reveals the direction perpendicular to the plane formed by \(\mathbf{b}\) and \(\mathbf{c}\).

In our example, \( \mathbf{b} = \langle -1, 4, 0 \rangle \) and \( \mathbf{c} = \langle 3, -1, 3 \rangle \) leads to the vector \(\langle 12, 3, -13\rangle\) as the cross product. This vector is crucial for finding the scalar triple product and assessing coplanarity.

Coplanarity of Vectors

Coplanarity refers to a set of vectors lying on the same plane. In three-dimensional space, we can determine if vectors are coplanar by examining the scalar triple product.
The scalar triple product involves three vectors, utilizing the cross product and dot product together: \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \). This operation results in a scalar value.

The condition for vectors to be coplanar is simple:

• If the scalar triple product equals zero, the vectors are coplanar. This means they lie on the same geometric plane.
• If the scalar triple product is not zero, the vectors are not coplanar, and they span three-dimensional space.

In the given problem, the scalar triple product of vectors \( \mathbf{a}, \mathbf{b}, \) and \( \mathbf{c} \) equals 59, which is non-zero. Therefore, these vectors are not coplanar. They do not all reside on a single plane.

Volume of Parallelepiped

A parallelepiped is a six-faced figure (also known as a polyhedron) with opposite faces being parallelograms. When three vectors originate from a single point, they form the edges of a parallelepiped in three-dimensional space.
The volume of this solid shape is given by the absolute value of the scalar triple product of the vectors \(\mathbf{a}, \mathbf{b},\) and \(\mathbf{c}\). This is an extension of using the cross product to calculate the area of a parallelogram in two-dimensional space.

For volume calculation:

• Compute the cross product, \( \mathbf{b} \times \mathbf{c} \), to find a vector perpendicular to vectors \(\mathbf{b}\) and \(\mathbf{c}\).
• Use the dot product of this result with vector \( \mathbf{a} \) to obtain the scalar triple product.
• The absolute value of this number gives you the volume. It’s the measure of space inside the parallelepiped.

In this exercise, the scalar triple product is calculated to be 59, therefore the volume of the parallelepiped formed by the vectors \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) is \( 59 \) cubic units.