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What is the Coplanarity of Four Points using Vectors?
Grade Level:
Class 10
AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine
Definition
What is it?
Coplanarity of four points means that all four points lie on the same flat surface, just like four coins placed flat on a table. Using vectors, we can check this by seeing if the volume formed by three vectors created from these points is zero.
Simple Example
Quick Example
Imagine your family is sitting around a rectangular dining table. If all four members (let's say A, B, C, D) are sitting on the chairs at the table, their heads are roughly at the same height and form a flat plane. So, their heads are coplanar. If one person stands up on their chair, their head is no longer on the same flat surface, and thus, the four heads are not coplanar.
Worked Example
Step-by-Step
Let's check if points A(1, 2, 3), B(4, 5, 6), C(7, 8, 9), and D(10, 11, 12) are coplanar.
Step 1: Choose one point as the origin for our vectors. Let's pick A.
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Step 2: Form three vectors using A as the starting point: vector AB, vector AC, and vector AD.
Vector AB = B - A = (4-1, 5-2, 6-3) = (3, 3, 3)
Vector AC = C - A = (7-1, 8-2, 9-3) = (6, 6, 6)
Vector AD = D - A = (10-1, 11-2, 12-3) = (9, 9, 9)
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Step 3: Calculate the scalar triple product of these three vectors. If the scalar triple product is zero, the points are coplanar.
The scalar triple product is given by the determinant of the matrix formed by the components of the vectors:
| 3 3 3 |
| 6 6 6 |
| 9 9 9 |
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Step 4: Calculate the determinant.
Determinant = 3 * (6*9 - 6*9) - 3 * (6*9 - 6*9) + 3 * (6*11 - 6*10) (Oops, typo in calculation, let's re-calculate using the correct determinant formula for the given vectors)
Determinant = 3 * (6*9 - 6*9) - 3 * (6*9 - 6*9) + 3 * (6*9 - 6*9)
Determinant = 3 * (54 - 54) - 3 * (54 - 54) + 3 * (54 - 54)
Determinant = 3 * 0 - 3 * 0 + 3 * 0
Determinant = 0 - 0 + 0 = 0
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Step 5: Since the scalar triple product (determinant) 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 helps in many advanced fields like computer graphics to create realistic 3D models and animations, or in engineering to design stable structures. Civil engineers use this to ensure building foundations are flat, and in space technology, ISRO scientists use it for satellite trajectory planning.
Common Mistakes
MISTAKE: Picking any three vectors without a common starting point. | CORRECTION: Always form three vectors from a single common starting point among the four given points (e.g., AB, AC, AD if A is chosen).
MISTAKE: Incorrectly calculating the components of the vectors (e.g., A-B instead of B-A). | CORRECTION: Remember vector PQ = Q - P (final point coordinates minus initial point coordinates).
MISTAKE: Making calculation errors while finding the determinant of the 3x3 matrix. | CORRECTION: Double-check each multiplication and subtraction step carefully. Practice determinant calculations.
Practice Questions
Try It Yourself
QUESTION: Are the points P(0,0,0), Q(1,0,0), R(0,1,0), and S(1,1,1) coplanar? | ANSWER: No
QUESTION: Find the value of 'k' if the points A(1,1,1), B(2,3,4), C(3,2,1) and D(4,k,5) are coplanar. | ANSWER: k = 4
QUESTION: A drone takes off from your terrace (point T). It flies to drop a parcel at your friend's house (F), then picks up a package from the market (M), and finally returns to a charging station (C). If T=(0,0,0), F=(2,3,1), M=(4,6,2), and C=(1,1,0), are the points T, F, M, and C coplanar? Show your working. | ANSWER: No, they are not coplanar.
MCQ
Quick Quiz
If four points are coplanar, what will be the scalar triple product of the three vectors formed from these points (with a common origin)?
1
-1
Any non-zero value
The Correct Answer Is:
C
For four points to be coplanar, the three vectors formed from a common origin must lie in the same plane. This means they cannot form a volume, so their scalar triple product (which represents the volume of the parallelepiped formed by them) must be zero.
Real World Connection
In the Real World
Imagine a construction site for a new building in Mumbai. Before laying the foundation, engineers use laser levels and surveying equipment to ensure that all four corners of the building's base lie on the exact same flat horizontal plane. This is a real-world application of checking for coplanarity to ensure a stable and level structure.
Key Vocabulary
Key Terms
VECTOR: A quantity having both magnitude and direction, like displacement or force. | COPLANAR: Lying on the same flat plane. | SCALAR TRIPLE PRODUCT: A mathematical operation involving three vectors that results in a scalar value, representing the volume of a parallelepiped. | DETERMINANT: A special number calculated from a square matrix, used in solving systems of linear equations and finding vector products.
What's Next
What to Learn Next
Great job understanding coplanarity! Next, you can explore 'Equation of a Plane in 3D Space' using vectors. This will show you how to describe the flat surface on which coplanar points lie, taking your understanding of 3D geometry even further!
