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

Step-by-Step Solution

Answer

The scalar triple product is 2. The vectors are not coplanar. Volume = 2.

Step 1: Calculate the Cross Product

First, calculate the cross product \( \mathbf{b} \times \mathbf{c} \). To do this, use the determinant matrix method. The components are:\[ \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ -1 & 0 & 1 \ 0 & -1 & 1 \end{vmatrix} \] Calculate each component:- \( \mathbf{i} \) component: \( 0(1) - 1(-1) = 1 \)- \( \mathbf{j} \) component: \( -1(1) - 1(0) = -1 \)- \( \mathbf{k} \) component: \( -1(-1) - 0 = 1 \)Thus, \( \mathbf{b} \times \mathbf{c} = \langle 1, -1, 1 \rangle \).

Step 2: Calculate the Scalar Triple Product

Now, find the scalar triple product \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \), which uses dot product:\[ \mathbf{a} \cdot \langle 1, -1, 1 \rangle = 1 \times 1 + (-1) \times (-1) + 0 \times 1 \]Calculate: - \( 1 \times 1 = 1 \) - \( (-1) \times (-1) = 1 \) - \( 0 \times 1 = 0 \) Add these values: \( 1 + 1 + 0 = 2 \)The scalar triple product is 2.

Step 3: Determine Coplanarity and Volume

Vectors \( \mathbf{a} \), \( \mathbf{b} \), and \( \mathbf{c} \) are coplanar if their scalar triple product equals zero. Since we calculated the scalar triple product as 2, the vectors are not coplanar.Thus, the volume of the parallelepiped is the absolute value of the scalar triple product, which is \( |2| = 2 \).

Key Concepts

Scalar Triple Product

The scalar triple product is a useful tool in vector calculus. It combines three vectors

• two of them forming a cross product
• and the third completing a dot product with the result.

If you have vectors \( \mathbf{a} \), \( \mathbf{b} \), and \( \mathbf{c} \), the scalar triple product is written as \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \). The scalar triple product can be evaluated using the determinant of a 3x3 matrix, which provides a scalar value.
The scalar value indicates the volume of the parallelepiped formed by the vectors. Here's a quick way to understand this:

• The absolute value of the scalar triple product gives you the volume of the parallelepiped.
• If the product equals zero, the vectors are coplanar, meaning they lie within the same plane.

This product is very helpful in determining coplanarity and calculating volumes in physics and engineering applications.

Cross Product

The cross product of two vectors yields another vector that is perpendicular to both. For vectors \( \mathbf{b} \) and \( \mathbf{c} \), the cross product is denoted as \( \mathbf{b} \times \mathbf{c} \).
To calculate this, we often use the determinant method with a 3x3 matrix like this:

• Put the unit vectors \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \) in the first row.
• Place the components of \( \mathbf{b} \) in the second row.
• Position the components of \( \mathbf{c} \) in the third row.

Compute the determinant to derive a vector in the form \( \langle x, y, z \rangle \). This result is perpendicular to both \( \mathbf{b} \) and \( \mathbf{c} \), known as the normal vector to the plane established by these vectors.

Coplanar Vectors

Coplanar vectors are those that lie on the same flat surface or plane. In the context of scalar triple products, vectors \( \mathbf{a} \), \( \mathbf{b} \), and \( \mathbf{c} \) are coplanar if:

• The scalar triple product \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \) is zero.

When vectors are coplanar, it means there is no '+height+' element to form a volume, hence a zero scalar triple product indicates they all lie within the same plane.
If they are not coplanar, the above product indicates the volume of the shadow container they form, known as a parallelepiped, thus verifying and using this test is a simple yet powerful method of determining spatial relationships of vectors.