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What is the Coplanarity of Four Points using Scalar Triple Product?
Grade Level:
Class 12
AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics
Definition
What is it?
Coplanarity of four points means that all four points lie on the same flat surface or plane, just like four marbles resting on a single table. We can check if four points are coplanar using the Scalar Triple Product (STP), which helps us find the volume formed by three vectors.
Simple Example
Quick Example
Imagine you have four friends, A, B, C, and D, standing in a field. If they are all standing on the same perfectly flat cricket pitch, they are coplanar. If one friend is on a small hill and the others are on the pitch, they are not coplanar.
Worked Example
Step-by-Step
Let's check if points A(1, 2, 3), B(2, 3, 4), C(3, 4, 5), and D(4, 5, 6) are coplanar.
Step 1: Form three vectors from these four points. Pick one point as the starting point, say A. Then find vectors AB, AC, and AD.
Vector AB = B - A = (2-1, 3-2, 4-3) = (1, 1, 1)
Vector AC = C - A = (3-1, 4-2, 5-3) = (2, 2, 2)
Vector AD = D - A = (4-1, 5-2, 6-3) = (3, 3, 3)
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Step 2: Calculate the Scalar Triple Product (STP) of these three vectors. The STP is given by the determinant of the matrix formed by the components of the vectors.
STP = [AB AC AD] = AB . (AC x AD)
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Step 3: Calculate the cross product AC x AD.
AC x AD = | i j k |
| 2 2 2 |
| 3 3 3 |
= i(2*3 - 2*3) - j(2*3 - 2*3) + k(2*3 - 2*3)
= i(6 - 6) - j(6 - 6) + k(6 - 6)
= 0i - 0j + 0k = (0, 0, 0)
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Step 4: Calculate the dot product of AB with (AC x AD).
AB . (AC x AD) = (1, 1, 1) . (0, 0, 0)
= (1*0) + (1*0) + (1*0)
= 0 + 0 + 0 = 0
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Step 5: If the Scalar Triple Product is zero, the vectors are coplanar. Since the STP is 0, the vectors AB, AC, and AD are coplanar. This means the points A, B, C, and D are coplanar.
Answer: The points A, B, C, and D are coplanar.
Why It Matters
Understanding coplanarity is super important in fields like engineering and computer graphics. Engineers use it to design stable structures like bridges and buildings, ensuring parts fit perfectly. In AI/ML, it helps in understanding data patterns in 3D space, which is useful for things like self-driving cars navigating roads.
Common Mistakes
MISTAKE: Students sometimes forget to choose one common starting point for all three vectors. | CORRECTION: Always choose one of the four given points (e.g., A) and form three vectors starting from it (AB, AC, AD).
MISTAKE: Errors in calculating the determinant (Scalar Triple Product), especially with negative signs. | CORRECTION: Practice determinant calculations carefully. Remember the sign pattern: + - + for the first row expansion.
MISTAKE: Confusing coplanarity with collinearity (points on a line). | CORRECTION: Coplanarity means points are on the same plane (flat surface). Collinearity means points are on the same straight line.
Practice Questions
Try It Yourself
QUESTION: Are the points P(0,0,0), Q(1,0,0), R(0,1,0), S(0,0,1) coplanar? | ANSWER: No, they are not coplanar.
QUESTION: Check if points A(1,1,1), B(2,2,2), C(3,3,3), D(4,4,4) are coplanar. | ANSWER: Yes, they are coplanar.
QUESTION: Given points E(1,2,3), F(4,5,6), G(7,8,9), and H(x,y,z). If E, F, G are coplanar, find the condition for H to also be coplanar with E, F, G. (Hint: What is special about E, F, G?) | ANSWER: E, F, G are collinear. For H to be coplanar with them, H must lie on the line passing through E, F, G, or on any plane containing that line. The condition is (x-1)/3 = (y-2)/3 = (z-3)/3.
MCQ
Quick Quiz
If four points are coplanar, what will be the value of the Scalar Triple Product of the three vectors formed by choosing one common initial point?
1
-1
Any non-zero value
The Correct Answer Is:
C
If four points are coplanar, the three vectors formed from them lie on the same plane. The Scalar Triple Product represents the volume of the parallelepiped formed by these vectors. If the vectors are coplanar, this volume is zero.
Real World Connection
In the Real World
Imagine a drone delivering a package in a city. Its navigation system uses vectors to plot its path. If it needs to fly in a straight line or follow a specific flat trajectory (like staying at a certain altitude above a flat roof), understanding coplanarity helps ensure it stays on the correct 'plane' of movement. This is similar to how ISRO engineers plan satellite orbits.
Key Vocabulary
Key Terms
Coplanar: Points lying on the same flat surface | Scalar Triple Product (STP): A mathematical operation yielding a scalar value, representing the volume of a parallelepiped | Vector: A quantity having both magnitude and direction | Determinant: A special number associated with a square matrix, used in calculating STP
What's Next
What to Learn Next
Great job understanding coplanarity! Next, you can explore 'Equation of a Plane in 3D'. This concept builds on coplanarity by showing how to mathematically describe the flat surface on which coplanar points lie, which is vital for advanced geometry and physics.
