We are tasked with proving that four points $A$, $B$, $C$, and $D$ are coplanar if and only if the scalar triple product $\mathbf{AD} \cdot (\mathbf{AB} \times \mathbf{BC}) = 0$.

Definitions and Key Concepts

• Let $\mathbf{A}, \mathbf{B}, \mathbf{C}, \mathbf{D}$ be the position vectors corresponding to the points $A$, $B$, $C$, and $D$, respectively.
• Vectors $\mathbf{AB} = \mathbf{B} - \mathbf{A}$, $\mathbf{BC} = \mathbf{C} - \mathbf{B}$, and $\mathbf{AD} = \mathbf{D} - \mathbf{A}$ are formed between the points.
• Coplanar Points: Four points $A$, $B$, $C$, and $D$ are coplanar if they all lie on the same plane. In vector terms, three vectors drawn from one point (here, $A$) to the other three points must be linearly dependent.

Scalar Triple Product

The scalar triple product of vectors $\mathbf{AB}$, $\mathbf{BC}$, and $\mathbf{AD}$ is given by:

$\mathbf{AD} \cdot (\mathbf{AB} \times \mathbf{BC})$

This quantity represents the volume of the parallelepiped formed by the three vectors. If the scalar triple product equals zero, the volume is zero, which means the three vectors are coplanar, implying that the four points $A$, $B$, $C$, and $D$ lie on the same plane.

Proof

1. If the points $A$, $B$, $C$, and $D$ are coplanar, then $\mathbf{AD} \cdot (\mathbf{AB} \times \mathbf{BC}) = 0$:

• When the points are coplanar, the vectors $\mathbf{AB}$, $\mathbf{BC}$, and $\mathbf{AD}$ are linearly dependent.
• In vector terms, linear dependence means there is a scalar relation among the vectors, which implies that the volume of the parallelepiped formed by these vectors is zero.
• The scalar triple product measures this volume, so:

$\mathbf{AD} \cdot (\mathbf{AB} \times \mathbf{BC}) = 0$

This confirms that the points are coplanar.

2. Only if $\mathbf{AD} \cdot (\mathbf{AB} \times \mathbf{BC}) = 0$, then the points $A$, $B$, $C$, and $D$ are coplanar:

• If the scalar triple product is zero, the volume of the parallelepiped formed by $\mathbf{AB}$, $\mathbf{BC}$, and $\mathbf{AD}$ is zero.
• Zero volume indicates that the three vectors are coplanar, which means the points $A$, $B$, $C$, and $D$ all lie on the same plane.

Thus, the scalar triple product being zero is both a necessary and sufficient condition for the points $A$, $B$, $C$, and $D$ to be coplanar.

Conclusion

We have proven that four points $A$, $B$, $C$, and $D$ are coplanar if and only if:

$\mathbf{AD} \cdot (\mathbf{AB} \times \mathbf{BC}) = 0$

This condition shows that the volume formed by the vectors between these points is zero, ensuring their coplanarity.

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Would you like more details or have any questions? Here are some related questions:

1. What are some real-world applications of coplanar points?
2. How do we compute the scalar triple product in practice?
3. Can you prove that vectors being linearly dependent implies coplanarity?
4. How does the scalar triple product relate to determinants?
5. What is the geometric interpretation of the cross product?

Tip: The scalar triple product gives the signed volume of a parallelepiped formed by three vectors.