Problem 32 - 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. $$ \mathbf{a}=\langle 1,-1,0\rangle, \quad \mathbf{b}=\langle- 1,0,1\rangle, \quad \mathbf{c}=\langle 0,-1,1\rangle $$

Step-by-Step Solution

Answer

The scalar triple product is 0. The vectors are coplanar, so they do not form a parallelepiped.

Step 1: Calculate Cross Product

To find the scalar triple product \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \), we first need to calculate the cross product \( \mathbf{b} \times \mathbf{c} \). Use the formula for the cross product of two vectors \( \mathbf{u} = \langle u_1, u_2, u_3 \rangle \) and \( \mathbf{v} = \langle v_1, v_2, v_3 \rangle \) which is \[(\mathbf{u} \times \mathbf{v})_i = (u_2v_3 - u_3v_2, u_3v_1 - u_1v_3, u_1v_2 - u_2v_1)\]. For \( \mathbf{b} = \langle -1, 0, 1 \rangle \) and \( \mathbf{c} = \langle 0, -1, 1 \rangle \), we have: \[\mathbf{b} \times \mathbf{c} = \langle (0 \cdot 1) - (1 \cdot (-1)), (1 \cdot 0) - (-1 \cdot 1), ((-1) \cdot (-1)) - (0 \cdot 0)\rangle = \langle 0 + 1, 0 + 1, 1 - 0 \rangle = \langle 1, 1, 1 \rangle\].

Step 2: Compute Scalar Triple Product

Now, calculate \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \). Using the dot product formula \( \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + u_3v_3 \) for \( \mathbf{a} = \langle 1, -1, 0 \rangle \) and \( \mathbf{b} \times \mathbf{c} = \langle 1, 1, 1 \rangle \), we have: \[ \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = (1 \cdot 1) + ((-1) \cdot 1) + (0 \cdot 1) = 1 - 1 + 0 = 0. \]

Step 3: Determine Coplanarity

Vectors \( \mathbf{a}, \mathbf{b}, \) and \( \mathbf{c} \) are coplanar if their scalar triple product is zero. Since we found that \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = 0 \), the vectors are coplanar. No volume calculation is necessary as they do not form a parallelepiped.

Key Concepts

Vector Cross Product

The concept of the vector cross product is fundamental in vector calculus and physics as it is used to compute the area and orientation of parallelograms defined by two vectors in three-dimensional space. The cross product of two vectors \( \mathbf{u} = \langle u_1, u_2, u_3 \rangle \) and \( \mathbf{v} = \langle v_1, v_2, v_3 \rangle \) results in a new vector that is perpendicular to both \( \mathbf{u} \) and \( \mathbf{v} \).
The formula for the cross product is given as:

• First component: \( u_2v_3 - u_3v_2 \)
• Second component: \( u_3v_1 - u_1v_3 \)
• Third component: \( u_1v_2 - u_2v_1 \)

This leads to\( \mathbf{u} \times \mathbf{v} = \langle u_2v_3 - u_3v_2, u_3v_1 - u_1v_3, u_1v_2 - u_2v_1 \rangle \). This resulting vector's magnitude is equal to the area of the parallelogram spanned by \( \mathbf{u} \) and \( \mathbf{v} \).

In the provided exercise, we demonstrated this by calculating \( \mathbf{b} \times \mathbf{c} \), which was \( \langle 1, 1, 1 \rangle \). This step is crucial for understanding subsequent operations, such as the scalar triple product.

Dot Product

The dot product, also known as the scalar product, is a foundational concept in vector mathematics. It calculates the sum of the products of the corresponding components of two vectors, resulting in a scalar.
The formula for the dot product of two vectors \( \mathbf{u} = \langle u_1, u_2, u_3 \rangle \) and \( \mathbf{v} = \langle v_1, v_2, v_3 \rangle \) is:

• \( \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + u_3v_3 \)

This product captures the degree to which two vectors point in the same direction. It is crucial in applications like calculating angles between vectors and in physical interpretations such as calculating work done by a force.

In the context of our exercise, the scalar triple product \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \) was calculated using the dot product. The result was 0, providing relevant information about the spatial relationship of the vectors involved.

Coplanar Vectors

Vectors are considered to be coplanar if they lie in the same plane. This means that there is no volume spanned by the vectors when seen together. A common test for coplanarity in three-dimensional space is through the scalar triple product.
If the scalar triple product of three vectors \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) is zero, i.e., \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = 0 \), the vectors are coplanar.

Coplanar vectors do not span any volume, as the zero result from the scalar triple product indicates the absence of a parallelepiped formed by the vectors. This concept is crucial in fields like computer graphics, physics, and engineering, where understanding spatial configurations is essential.

In the exercise, since the scalar triple product resulted in zero, it confirmed that the vectors \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) are indeed coplanar. Recognizing coplanarity helps in simplifying problems and computations in vector spaces.