Problem 38 - Essential Calculus Early Transcendentals

Question

Use the scalar triple product to determine whether the points \(A(1,3,2), B(3,-1,6), C(5,2,0),\) and \(D(3,6,-4)\) lie in the same plane.

Step-by-Step Solution

Answer

The points lie in the same plane because the scalar triple product is zero.

Step 1: Understand the Scalar Triple Product

The scalar triple product is useful for determining if points are coplanar. It combines three vectors and can be calculated as \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \). If it equals zero, the vectors (and points they originate from) are coplanar.

Step 2: Find the Vectors

First, calculate the vectors based on the given points. Let \( \mathbf{AB} = \mathbf{B} - \mathbf{A} = (3-1, -1-3, 6-2) = (2, -4, 4) \), \( \mathbf{AC} = \mathbf{C} - \mathbf{A} = (5-1, 2-3, 0-2) = (4, -1, -2) \), and \( \mathbf{AD} = \mathbf{D} - \mathbf{A} = (3-1, 6-3, -4-2) = (2, 3, -6) \).

Step 3: Compute the Cross Product \(\mathbf{AC} \times \mathbf{AD}\)

Use the formula for the cross product: \( \mathbf{u} \times \mathbf{v} = (u_2v_3 - u_3v_2, u_3v_1 - u_1v_3, u_1v_2 - u_2v_1) \). Applying this to \( \mathbf{AC} \) and \( \mathbf{AD} \):\[ \mathbf{AC} \times \mathbf{AD} = ((-1)(-6) - (-2)(3), (-2)(2) - 4(-6), 4(3) - (-1)(2)) \] Calculating each component:\( \mathbf{AC} \times \mathbf{AD} = (6 + 6, -4 + 24, 12 + 2) = (12, 20, 14) \).

Step 4: Calculate the Scalar Triple Product \( \mathbf{AB} \cdot (\mathbf{AC} \times \mathbf{AD}) \)

Compute the dot product using \( \mathbf{AB} = (2, -4, 4) \) and \( \mathbf{AC} \times \mathbf{AD} = (12, 20, 14) \):\[ \mathbf{AB} \cdot (\mathbf{AC} \times \mathbf{AD}) = 2 \times 12 + (-4) \times 20 + 4 \times 14 = 24 - 80 + 56 \] Calculate:\( 24 - 80 + 56 = 0 \).

Step 5: Conclusion on Coplanarity

Since the scalar triple product equals zero, the vectors \( \mathbf{AB} \), \( \mathbf{AC} \), and \( \mathbf{AD} \) are coplanar. This means that the points \( A, B, C, \/ \text{and} \/ D \) lie in the same plane.

Key Concepts

Coplanarity

Coplanarity refers to the condition when several points or vectors lie on the same plane in a three-dimensional space. In simpler terms, imagine being able to stretch a flat sheet through all these points without lifting the sheet at any point. For points to be coplanar, the scalar triple product must equate to zero.
The scalar triple product formula can be expressed as \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \), where \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) are vectors derived from the points. If the computed result is zero, this confirms that the points are indeed coplanar.

• Important Note: The scalar triple product is invaluable because it gives a straightforward method to verify coplanarity without complex calculations.

In the given problem, we calculated vectors from the points and found the scalar triple product. Since it turned out to be zero, we confirmed that the points A, B, C, and D lie on the same plane. This illustrates how coplanarity is crucial for understanding the spatial relationship between points.

Vector Cross Product

The vector cross product is another fundamental concept in vector calculus, which helps us find a vector that is perpendicular to two given vectors in three-dimensional space. If you imagine two vectors throwing shadows on the floor, the cross product will give the direction of a stick poking upward, perpendicular to the plane laid out by these vectors.
This is incredibly helpful when determining spatial relationships or performing calculations involving three-dimensional vectors.

• Calculation: The cross product of two vectors \( \mathbf{u} \) and \( \mathbf{v} \) is calculated as:
  \[ \mathbf{u} \times \mathbf{v} = (u_2v_3 - u_3v_2, u_3v_1 - u_1v_3, u_1v_2 - u_2v_1) \]

In the exercise, we found the cross product \( \mathbf{AC} \times \mathbf{AD} \), resulting in \( (12, 20, 14) \). Calculating the cross product is the crucial step before determining the scalar triple product, as it forms part of further calculations. The result tells us about the directional relationship between the original vectors \( \mathbf{AC} \) and \( \mathbf{AD} \).

Dot Product

The dot product contributes significantly to understanding vector relationships. It computes a scalar (a single number) from two vectors, representing how much one vector extends in the direction of another. In this task, it determines the parallelism or perpendicularity of vectors in terms of multiplication, summarizing their interaction.

• Formula: The dot product for vectors \( \mathbf{u} = (u_1, u_2, u_3) \) and \( \mathbf{v} = (v_1, v_2, v_3) \) is given by:
  \( \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + u_3v_3 \)

In the original problem, we calculated the dot product between vector \( \mathbf{AB} \) and the cross product \( \mathbf{AC} \times \mathbf{AD} \). It resulted in zero, which confirmed the coplanarity of the points. This zero value indicates that these vectors are orthogonal (at right angles) to the plane formed by \( \mathbf{AC} \) and \( \mathbf{AD} \), supporting our conclusion of coplanarity.