Problem 31 - Algebra and Trigonometry

Question

Three vectors \(\mathbf{a}, \mathbf{b},\) and \(\mathbf{c}\) are given. (a) Find their scalar triple product \(\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c}) .\) (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, and the vectors are not coplanar. The volume of the parallelepiped is 59 cubic units.

Step 1: Calculate the Cross Product

To find the scalar triple product \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \), we first need to calculate the cross product \( \mathbf{b} \times \mathbf{c} \). Using the formula for cross product: \( \mathbf{b} \times \mathbf{c} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ -1 & 4 & 0 \ 3 & -1 & 3 \end{vmatrix} \),expand to solve: \( = \mathbf{i}(4 \cdot 3 - 0 \cdot (-1)) - \mathbf{j}(-1 \cdot 3 - 0 \cdot 3) + \mathbf{k}(-1 \cdot (-1) - 4 \cdot 3) \), \( = \mathbf{i}(12) - \mathbf{j}(-3) + \mathbf{k}(-1 - 12) \), \( = \langle 12, 3, -13 \rangle \).

Step 2: Calculate the Scalar Triple Product

Now calculate \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \) using \( \mathbf{a} = \langle 2, 3, -2 \rangle \) and \( (\mathbf{b} \times \mathbf{c}) = \langle 12, 3, -13 \rangle \). Use the dot product formula: \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = 2 \cdot 12 + 3 \cdot 3 + (-2) \cdot (-13) \). Calculate: \( 24 + 9 + 26 = 59 \).

Step 3: Determine Coplanarity and Volume

The vectors are coplanar if the scalar triple product is zero. Since \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = 59 eq 0 \), the vectors are not coplanar. This non-zero value also represents the volume of the parallelepiped formed by the vectors. Therefore, the volume is 59 cubic units.

Key Concepts

Scalar Triple Product

The scalar triple product is a crucial operation that involves three vectors: \( \mathbf{a}, \mathbf{b} \), and \( \mathbf{c} \). It is denoted as \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \). This operation involves two key vector operations: the cross product and the dot product.

• First, calculate the cross product of two vectors, say \( \mathbf{b} \) and \( \mathbf{c} \).
• Then, take the dot product of the result with the third vector \( \mathbf{a} \).

This combined operation yields a scalar value, representing some important geometric properties. Specifically, it relates to whether the vectors are coplanar and, if they form a three-dimensional shape, its volume.

Cross Product

The cross product, or vector product, is a binary operation on two vectors in three-dimensional space. For vectors \( \mathbf{b} = \langle -1,4,0 \rangle \) and \( \mathbf{c} = \langle 3,-1,3 \rangle \), the cross product is found using a determinant:\[\mathbf{b} \times \mathbf{c} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ -1 & 4 & 0 \ 3 & -1 & 3 \end{vmatrix}\]This determinant involves breaking down the components and leads to the vector \( \langle 12, 3, -13 \rangle \). The process includes:

• Calculating the components by performing small sub-determinants for each unit vector \( \mathbf{i}, \mathbf{j}, \mathbf{k} \).
• Following the pattern to eliminate rows and columns associated with each unit vector.

The result is another vector that is perpendicular to both original vectors \( \mathbf{b} \) and \( \mathbf{c} \).

Dot Product

The dot product is a straightforward operation to combine two vectors and it yields a scalar. In the context of the scalar triple product, it serves as the second operation to compute the final value. Using our vectors \( \mathbf{a} = \langle 2,3,-2 \rangle \) and the cross product result \( \mathbf{b} \times \mathbf{c} = \langle 12, 3, -13 \rangle \), we calculate the dot product:\[\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = 2 \cdot 12 + 3 \cdot 3 + (-2) \cdot (-13)\]Breaking it down into smaller parts:

• \( 2 \times 12 = 24 \)
• \( 3 \times 3 = 9 \)
• \( -2 \times -13 = 26 \)

Adding these results gives us \( 24 + 9 + 26 = 59 \). This value, 59, is central to understanding the geometric properties of the vectors.

Coplanarity

Coplanarity in vectors refers to whether three vectors lie on the same plane. For three vectors to be coplanar, their scalar triple product must be zero. Mathematically, if \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = 0 \), then the vectors are coplanar.However, in our solution:

• The scalar triple product value is 59, not zero.

This result indicates that the vectors are not coplanar, meaning they span a three-dimensional space and form a volume.

Parallelepiped Volume

The volume of a parallelepiped is determined using the scalar triple product of three vectors. If the vectors are \( \mathbf{a}, \mathbf{b}, \mathbf{c} \), the volume is the absolute value of the scalar triple product, \( |\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) | \).

• The scalar triple product captures the {linear independence} needed to form a three-dimensional shape.
• Calculations reveal that the volume is 59 cubic units, based on our earlier result.

This value confirms that the vectors form a full three-dimensional structure rather than lying flat in a plane. The formula, therefore, provides a direct method to find the volume of the solid defined by these vectors.