Problem 37 - Essential Calculus Early Transcendentals

Question

Use the scalar triple product to verify that the vectors \(\mathbf{u}=\mathbf{i}+5 \mathbf{j}-2 \mathbf{k}, \mathbf{v}=3 \mathbf{i}-\mathbf{j},\) and \(\mathbf{w}=5 \mathbf{i}+9 \mathbf{j}-4 \mathbf{k}\) are coplanar.

Step-by-Step Solution

Answer

The vectors are not coplanar because their scalar triple product is -120, not zero.

Step 1: Understand the Scalars

The scalar triple product for vectors \( \mathbf{u}, \mathbf{v}, \mathbf{w} \) is given by the formula \( \mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) \). This value is the volume of the parallelepiped formed by the vectors. If the scalar triple product is zero, it indicates that the vectors are coplanar.

Step 2: Calculate the Cross Product \( \mathbf{v} \times \mathbf{w} \)

To find the cross product \( \mathbf{v} \times \mathbf{w} \), set up the determinant:\[ \mathbf{v} \times \mathbf{w} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ 3 & -1 & 0 \ 5 & 9 & -4 \end{vmatrix} \]Calculate it:- \( \mathbf{i} \) component: \((-1)(-4) - (0)(9) = 4\)- \( \mathbf{j} \) component: \(- ((3)(-4) - (0)(5)) = 12\)- \( \mathbf{k} \) component: \((3)(9) - ((-1)(5)) = 32\)So, \( \mathbf{v} \times \mathbf{w} = 4 \mathbf{i} - 12 \mathbf{j} + 32 \mathbf{k} \).

Step 3: Calculate the Dot Product \( \mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) \)

Now, compute the dot product \( \mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) \): \[ \mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) = (1)(4) + (5)(-12) + (-2)(32) \]Calculate each term and sum them:- \( 1 \cdot 4 = 4 \)- \( 5 \cdot (-12) = -60 \)- \( -2 \cdot 32 = -64 \)Thus, the sum \( 4 - 60 - 64 = -120 \).

Step 4: Conclusion on Coplanarity

Since the scalar triple product \( \mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) = -120 \), which is not zero, the vectors \( \mathbf{u}, \mathbf{v}, \mathbf{w} \) are not coplanar.

Key Concepts

Coplanar Vectors

Vectors are considered coplanar when they reside in the same geometric plane. For three vectors \(\mathbf{u}, \mathbf{v},\) and \( \mathbf{w} \), determining coplanarity involves calculating their scalar triple product.
The scalar triple product is expressed as \( \mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) \). If this product is zero, it means the vectors do not form a three-dimensional space and hence are coplanar.
This is because the scalar triple product represents the volume of the parallelepiped constructed by the vectors. A zero volume indicates a collapse into a plane, confirming that the vectors share a common plane.

• The concept relies heavily on vector algebra, particularly the cross and dot products.
• Geometrically, this scenario is akin to comparing the lengths of shadows on a flat surface.

Understanding coplanarity is crucial in physics and engineering, where spatial relationships govern material behavior.

Cross Product

The cross product, also known as the vector product, is a binary operation on two vectors in three-dimensional space. It is denoted by \( \mathbf{v} \times \mathbf{w} \) and results in a vector that is perpendicular to the plane formed by \( \mathbf{v} \) and \( \mathbf{w} \).
The magnitude of the cross product vector is equal to the area of the parallelogram that the vectors span. This property makes the cross product valuable in physics for determining torque, rotational forces, and more.

• To compute \( \mathbf{v} \times \mathbf{w} \), construct a 3x3 matrix determinant with the unit vectors \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) in the first row and the components of \( \mathbf{v} \) and \( \mathbf{w} \) in the second and third rows, respectively.
• Expand the determinant to find each component of the resulting vector.

In our exercise, we did this to find \( 4 \mathbf{i} - 12 \mathbf{j} + 32 \mathbf{k} \), which is critical for further calculations related to coplanarity.

Dot Product

The dot product, or scalar product, is an operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. Defined as \( \mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) \), it measures how much one vector "projects" onto another vector, resulting in a scalar.
This operation is essential in various applications, including the assessment of angle relationships or determining the work done by a force along a vector direction.

• The basic formula is \( \,a_1b_1 + a_2b_2 + a_3b_3\), where \(a_i\) and \(b_i\) represent the coordinates of vectors \(\mathbf{a}\) and \(\mathbf{b}\), respectively.
• In context, the dot product allows us to verify if the vectors are coplanar by realizing that the zero value marks orthogonality to the plane.

For our given vectors, the resulting product was \(-120\), indicating that they do not lie on a single plane, as this value was not zero.