Problem 30 - 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. $$ \mathbf{a}=\langle 3,0,-4\rangle, \quad \mathbf{b}=\langle 1,1,1\rangle, \quad \mathbf{c}=\langle 7,4,0\rangle $$

Step-by-Step Solution

Answer

The vectors are coplanar, and the volume is 0.

Step 1: Find the Cross Product \( \mathbf{b} \times \mathbf{c} \)

To find the cross product \( \mathbf{b} \times \mathbf{c} \), use the determinant method. Set up a 3x3 matrix with the unit vectors \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) in the first row, components of \( \mathbf{b} \) in the second row, and components of \( \mathbf{c} \) in the third row.\[\mathbf{b} \times \mathbf{c} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ 1 & 1 & 1 \ 7 & 4 & 0 \end{vmatrix}\]Now compute this determinant:

Step 2: Solve the Determinant for \( \mathbf{b} \times \mathbf{c} \)

Compute the determinant:\[\mathbf{b} \times \mathbf{c} = \mathbf{i}(1 \cdot 0 - 1 \cdot 4) - \mathbf{j}(1 \cdot 0 - 1 \cdot 7) + \mathbf{k}(1 \cdot 4 - 1 \cdot 7)\]\[= -4\mathbf{i} + 7\mathbf{j} -3\mathbf{k}\]So, \( \mathbf{b} \times \mathbf{c} = \langle -4, 7, -3 \rangle \).

Step 3: Calculate the Scalar Triple Product \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \)

Now use \( \mathbf{a} = \langle 3, 0, -4 \rangle \) and \( \mathbf{b} \times \mathbf{c} = \langle -4, 7, -3 \rangle \) to find the scalar triple product:\[\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = 3(-4) + 0(7) + (-4)(-3)\]\[= -12 + 0 + 12 = 0\]

Step 4: Determine if Vectors are Coplanar

If the scalar triple product is zero, the vectors are coplanar. We calculated \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = 0 \), so the vectors are coplanar.

Step 5: Conclude the Volume of the Parallelepiped

Since the vectors are coplanar, the volume of the parallelepiped is zero, as three coplanar vectors span no volume in space.

Key Concepts

Coplanar Vectors

In geometry, understanding if vectors are coplanar involves imagining how these vectors align in three-dimensional space. Vectors are coplanar if they lie on the same plane. If you think of three vectors as forming a triangle or a loop, if they all lie flat on a piece of paper, they are coplanar. This is a crucial observation for problems involving three-dimensional space. Mathematically, to determine if vectors are coplanar, one effective method is to calculate their **scalar triple product**. The scalar triple product of vectors **a**, **b**, and **c** is defined as \[\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})\]If this result is zero, the vectors do not span a volume, hence they must be coplanar. For example, with vectors \( \mathbf{a} = \langle 3, 0, -4 \rangle \), \( \mathbf{b} = \langle 1, 1, 1 \rangle \), and \( \mathbf{c} = \langle 7, 4, 0 \rangle \), by calculating their scalar triple product and getting zero, we conclude they are coplanar.

Cross Product

A cross product, often used in vector analysis, is a binary operation on two vectors in three-dimensional space. Imagine you have two vectors that you want to multiply to get another vector that is perpendicular to both of them. The cross product allows you to achieve this. To find the cross product \( \mathbf{b} \times \mathbf{c} \), you can follow these steps:

• Set up a 3x3 matrix with unit vectors in the first row.
• Place the components of \( \mathbf{b} \) in the second row and \( \mathbf{c} \) in the third row.

Then, compute the determinant of this matrix:\[\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \1 & 1 & 1 \7 & 4 & 0 \end{vmatrix}\]Solving this determinant, you find that:\(\mathbf{b} \times \mathbf{c} = \langle -4, 7, -3 \rangle\)The resulting vector is perpendicular to both \( \mathbf{b} \) and \( \mathbf{c} \). This characteristic makes the cross product especially useful in applications like physics and engineering.

Parallelepiped Volume

A parallelepiped is a three-dimensional shape formed by six parallelograms. Think of it as a skewed box. Calculating its volume is especially relevant when working with vector analysis where such a shape might be defined by three vectors. The volume of a parallelepiped constructed by vectors \( \mathbf{a} \), \( \mathbf{b} \), and \( \mathbf{c} \) can be found using the magnitude of the scalar triple product:\[V = |\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})|\]If you get a scalar triple product equal to zero, like in our example, this means the shape has no volume. This happens because the vectors are coplanar, indicating that these vectors lie in the same plane and thus cannot form a three-dimensional space.In practical applications, knowing how to compute the volume of a parallelepiped helps you understand the spatial relationships between vectors, which is crucial in many fields like computer graphics and physics for modeling and simulations.