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What is the Coplanarity Test for Four Points using Vectors?
Grade Level:
Class 12
AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics
Definition
What is it?
The Coplanarity Test for Four Points using Vectors helps us check if four given points lie on the same flat surface, like a tabletop. If they do, we say they are 'coplanar'. We use vector calculations to find this out.
Simple Example
Quick Example
Imagine you have four small coins on your study table. If all four coins are perfectly flat on the table, they are coplanar. But if one coin is tilted or slightly lifted, they are not coplanar. This test helps us mathematically figure out if they are all on the same 'flat surface' in space.
Worked Example
Step-by-Step
Let's check if points A(1, 2, 3), B(2, 3, 4), C(3, 4, 5), and D(1, 1, 1) are coplanar.
Step 1: Form three vectors using these points. Let's choose A as the common starting point.
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 = (1-1, 1-2, 1-3) = (0, -1, -2)
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Step 2: Calculate the scalar triple product of these three vectors. If the points are coplanar, this product will be zero. The scalar triple product is given by AB . (AC x AD).
First, find the cross product AC x AD:
AC x AD = | i j k |
| 2 2 2 |
| 0 -1 -2 |
= i * (2*-2 - 2*-1) - j * (2*-2 - 2*0) + k * (2*-1 - 2*0)
= i * (-4 + 2) - j * (-4 - 0) + k * (-2 - 0)
= -2i + 4j - 2k
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Step 3: Now, find the dot product of AB with (AC x AD):
AB . (AC x AD) = (1, 1, 1) . (-2, 4, -2)
= (1 * -2) + (1 * 4) + (1 * -2)
= -2 + 4 - 2
= 0
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Step 4: Since the scalar triple product is 0, the three vectors AB, AC, and AD are coplanar. This means the four points A, B, C, and D are also coplanar.
Answer: The points A(1, 2, 3), B(2, 3, 4), C(3, 4, 5), and D(1, 1, 1) are coplanar.
Why It Matters
Understanding coplanarity is super important in fields like Engineering and Computer Graphics. For example, in designing car parts or buildings, engineers use this to ensure surfaces fit perfectly. In creating movie special effects, graphic designers use it to make virtual objects look realistic, saving time and money!
Common Mistakes
MISTAKE: Not choosing a common starting point for all three vectors. | CORRECTION: Always choose one of the four given points (e.g., point A) as the starting point for all three vectors (AB, AC, AD).
MISTAKE: Making calculation errors in the cross product or dot product. | CORRECTION: Double-check your arithmetic, especially with negative signs, when calculating the determinant for the cross product and then the dot product.
MISTAKE: Forgetting that the scalar triple product must be zero for coplanarity. | CORRECTION: Remember, if the calculated scalar triple product is not zero, the points are NOT coplanar.
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: Determine if the points A(1,0,1), B(2,1,2), C(3,2,3), and D(0,-1,0) are coplanar. | ANSWER: Yes
QUESTION: Find the value of 'k' such that the points E(1,1,1), F(2,3,4), G(3,2,1), and H(4,k,5) are coplanar. | ANSWER: k = 4
MCQ
Quick Quiz
If four points A, B, C, and D are coplanar, what will be the value of the scalar triple product of vectors AB, AC, and AD?
1
-1
Any non-zero number
The Correct Answer Is:
B
For four points to be coplanar, the three vectors formed from a common origin among them must also be coplanar. The scalar triple product of three coplanar vectors is always zero.
Real World Connection
In the Real World
Imagine you're building a drone for delivering packages in a city like Bengaluru. The drone's flight path and the landing platform need to be precisely aligned. Engineers at companies like Zomato or Swiggy use principles of coplanarity to ensure the drone's cameras, sensors, and the package drop mechanism are all in the same plane for accurate delivery, making sure your food or groceries reach you safely.
Key Vocabulary
Key Terms
VECTOR: A quantity having both magnitude and direction | COPLANAR: Lying in the same plane | SCALAR TRIPLE PRODUCT: The dot product of one vector with the cross product of two other vectors | CROSS PRODUCT: A vector operation that results in a vector perpendicular to the two original vectors | DOT PRODUCT: A scalar operation that results in a scalar value
What's Next
What to Learn Next
Great job understanding coplanarity! Next, you can explore 'Equation of a Plane in 3D Space'. This will help you understand how to mathematically describe the 'flat surface' itself, which is a natural extension of knowing if points lie on it.
