Problem 34 - 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. $$ \mathbf{a}=2 \mathbf{i}-2 \mathbf{j}-3 \mathbf{k}, \quad \mathbf{b}=3 \mathbf{i}-\mathbf{j}-\mathbf{k}, \quad \mathbf{c}=6 \mathbf{i} $$

Step-by-Step Solution

Answer

(a) Scalar triple product is 18. (b) Vectors are not coplanar; the volume is 18.

Step 1: Understanding Scalar Triple Product

The scalar triple product of vectors \( \mathbf{a}, \mathbf{b}, \text{ and } \mathbf{c} \) is defined as \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \). It results in a scalar value.

Step 2: Calculate Cross Product of \( \mathbf{b} \times \mathbf{c} \)

To find \( \mathbf{b} \times \mathbf{c} \), use the determinant formula for a cross product, where the vectors are placed in a 3x3 determinant with unit vectors \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) as the first row:\[\begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \3 & -1 & -1 \6 & 0 & 0\end{vmatrix}\]Calculate the determinant by expanding along the first row:\[\mathbf{i}((-1)(0) - (-1)(0)) - \mathbf{j}(3(0) - (-1)(6)) + \mathbf{k}(3\times0 - (-1)\times6)\]This simplifies to \( 0 \mathbf{i} - 18 \mathbf{j} + 6 \mathbf{k} \).

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

Substitute the result of \( \mathbf{b} \times \mathbf{c} = 0\mathbf{i} - 18\mathbf{j} + 6\mathbf{k} \) into \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \) and calculate:\[(2\mathbf{i} - 2\mathbf{j} - 3\mathbf{k}) \cdot (0\mathbf{i} - 18\mathbf{j} + 6\mathbf{k})\]\[= 2\times0 + (-2)(-18) + (-3)(6)\]\[= 0 + 36 - 18 = 18\].

Step 4: Determine if the Vectors are Coplanar

Vectors are coplanar if their scalar triple product is zero. Because \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = 18 \), the vectors are not coplanar.

Step 5: Calculate the Volume of the Parallelepiped

The absolute value of the scalar triple product gives the volume of the parallelepiped formed by the vectors. Therefore, the volume is \(|18| = 18\).

Key Concepts

Coplanar Vectors

Vectors are considered coplanar when they lie in the same geometric plane. In simpler terms, if you can arrange the vectors so they all rest on a single flat surface with no overlap, they are coplanar.
In mathematics, checking if vectors are coplanar can be done using the scalar triple product. If three vectors \( \mathbf{a}, \mathbf{b}, \) and \( \mathbf{c} \) are given, their scalar triple product is given as \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \).
The key thing to look for is whether this product is zero or not.

• If the scalar triple product is zero, the vectors are coplanar.
• If it's not zero, the vectors are not coplanar.

In the given exercise, the scalar triple product turned out to be 18. Since it's not zero, the vectors are not coplanar. This means there is no single plane that can accommodate all three vectors simultaneously.

Cross Product

The cross product is a fundamental operation in vector mathematics, particularly in 3-dimensional space. It involves two vectors and results in a third vector that is perpendicular to the plane formed by the original two vectors.
The formula includes calculating the determinant of a 3x3 matrix with the unit vectors \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) in the first row, and the components of the two vectors in the next two rows. In our exercise, to calculate the cross product \( \mathbf{b} \times \mathbf{c} \), we used the determinant method as follows:
\[\begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \ 3 & -1 & -1 \ 6 & 0 & 0\end{vmatrix}\]
By expanding along the first row, we derive the components of the resulting vector. The cross product simplifies to \( 0 \mathbf{i} - 18 \mathbf{j} + 6 \mathbf{k} \). This vector lies perpendicular to the plane formed by vectors \( \mathbf{b} \) and \( \mathbf{c} \).

Volume of Parallelepiped

A parallelepiped is a 3-dimensional figure formed by six parallelograms. If three vectors \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) define the edges meeting at one vertex of a parallelepiped, the volume can be determined using the scalar triple product.
The concept is fairly straightforward:

• The volume is equal to the absolute value of the scalar triple product of the three vectors.
• This product gives a measure of how much space the parallelepiped occupies in 3-dimensional space.

In our problem, the scalar triple product was calculated to be 18. Thus, the volume of the parallelepiped formed by vectors \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) is |18|, which simplifies to 18 cubic units. This is an important concept in vector mathematics as it gives a clear understanding of the spatial properties of the vectors involved.