Problem 55 - University Calculus: Early Transcendentals

Question

Determine whether the given points are coplanar. $$A(1,1,1), \quad B(-1,0,4), \quad C(0,2,1), \quad D(2,-2,3)$$

Step-by-Step Solution

Answer

The points are coplanar.

Step 1: Set up the position vectors

Start by defining the position vectors of the points with respect to point A. Let \( \mathbf{AB} = B - A \), \( \mathbf{AC} = C - A \), and \( \mathbf{AD} = D - A \). This gives us:- \( \mathbf{AB} = (-1 - 1, 0 - 1, 4 - 1) = (-2, -1, 3) \)- \( \mathbf{AC} = (0 - 1, 2 - 1, 1 - 1) = (-1, 1, 0) \)- \( \mathbf{AD} = (2 - 1, -2 - 1, 3 - 1) = (1, -3, 2) \)

Step 2: Calculate the cross product of two vectors

Calculate the cross product \( \mathbf{AB} \times \mathbf{AC} \) to find a vector perpendicular to both. The formula for the cross product is given by\[ \mathbf{AB} \times \mathbf{AC} = (y_1z_2 - z_1y_2, z_1x_2 - x_1z_2, x_1y_2 - y_1x_2) \].Substitute the values:- \( y_1 = -1, z_1 = 3, x_1 = -2 \)- \( x_2 = -1, y_2 = 1, z_2 = 0 \)This yields the cross product:\[ \mathbf{AB} \times \mathbf{AC} = ((-1) \cdot 0 - 3 \cdot 1, 3 \cdot (-1) - (-2) \cdot 0, (-2) \cdot 1 - (-1) \cdot (-1)) \]\[ = (0 - 3, -3 - 0, -2 - 1) = (-3, -3, -3) \].

Step 3: Calculate the dot product with the third vector

Calculate the dot product of the result from Step 2 with the vector \( \mathbf{AD} \). If the dot product is zero, the points are coplanar.\[ \mathbf{n} \cdot \mathbf{AD} = (-3, -3, -3) \cdot (1, -3, 2) = (-3 \cdot 1) + (-3 \cdot -3) + (-3 \cdot 2) \]\[ = -3 + 9 - 6 = 0 \].

Step 4: Conclude about coplanarity

Since the dot product is zero, the vector \( \mathbf{AD} \) is perpendicular to the cross product of \( \mathbf{AB} \) and \( \mathbf{AC} \). This indicates that the four points \( A, B, C, D \) are coplanar. This confirms that the volume of the parallelepiped defined by these vectors is zero.

Key Concepts

Position Vectors

Position vectors are crucial in understanding relationships between points in space. They allow us to describe a point's location relative to another reference point.
In this problem, each position vector starts from the fixed point \( A \) and points towards \( B \), \( C \), or \( D \).

• For vector \( \mathbf{AB} \), the components are calculated by subtracting the coordinates of \( A \) from \( B \). This results in \( \mathbf{AB} = (-2, -1, 3) \).
• Similarly, \( \mathbf{AC} \) and \( \mathbf{AD} \) are derived, providing vector directions from \( A \) to those points.

Understanding position vectors is key to transitioning into vector operations, such as the cross and dot product.

Cross Product

The cross product is a fundamental operation in vector mathematics, particularly useful for determining perpendicular vectors. This operation applies to two vectors in three-dimensional space.
The result is another vector, referred to as \( \mathbf{n} \), which is perpendicular to both given vectors.In this problem, we calculate the cross product of \( \mathbf{AB} \) and \( \mathbf{AC} \) to obtain a perpendicular vector. The formula used is \[ \mathbf{AB} \times \mathbf{AC} = (y_1z_2 - z_1y_2, z_1x_2 - x_1z_2, x_1y_2 - y_1x_2) \]By inserting the respective components of the vectors, we find the cross product \( \mathbf{AB} \times \mathbf{AC} = (-3, -3, -3) \). This vector is crucial for assessing coplanarity.

Dot Product

The dot product is an operation that evaluates the angle between two vectors, providing insight into their directional relationship.
In this problem, the dot product between \( \mathbf{n} \) and \( \mathbf{AD} \) helps determine coplanarity.

• If the result of the dot product is zero, it signifies that the vectors are perpendicular.
• Such perpendicularity implies that the four points \( A, B, C, \) and \( D \) lie in the same plane, as shown by \( \mathbf{n} \cdot \mathbf{AD} = 0 \).

Understanding how the dot product works simplifies handling spatial relationships in vector mathematics.

Vector Mathematics

Vector mathematics is a versatile toolset used to describe and solve spatial problems in geometry and physics.
With the current exercise, we rely heavily on operations like cross and dot products to verify coplanarity of points. Vectors provide an efficient way to handle multi-dimensional data. Some key components include:

• Addition and subtraction: Combining vectors allows for translating between different points in space.
• Cross product: Determines perpendicularity, essential in many physical and geometric problems.
• Dot product: Helps in ascertaining angles and orthogonality between vectors.

By mastering these concepts, students can approach a variety of vector-related problems with confidence.