What is the Condition for Coplanarity? | Simple Explanation for Class 12

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What is the Condition for Coplanarity using Vectors?

Grade Level:

Class 12

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Definition

What is it?

For three vectors to be coplanar, it means they all lie on the same flat surface or plane. The condition for this using vectors is that their scalar triple product must be zero. This scalar triple product is essentially the volume of the parallelepiped formed by the three vectors; if they are coplanar, this volume is zero.

Simple Example

Quick Example

Imagine three cricket bats. If you lay all three bats flat on the ground, they are coplanar. If you try to make one bat stand up while the other two are flat, they are not coplanar. In vector terms, if the 'box' they form has zero volume, they are coplanar.

Worked Example

Step-by-Step

PROBLEM: Check if the vectors a = i - 2j + k, b = 2i + j - k, and c = 7i - 4j + k are coplanar.

STEP 1: Write down the vectors in component form.
a = (1, -2, 1)
b = (2, 1, -1)
c = (7, -4, 1)
---STEP 2: Calculate the scalar triple product, which is the determinant of the matrix formed by their components.
Scalar Triple Product = a . (b x c)
This is calculated as the determinant:
| 1 -2 1 |
| 2 1 -1 |
| 7 -4 1 |
---STEP 3: Expand the determinant.
= 1 * ((1 * 1) - (-1 * -4)) - (-2) * ((2 * 1) - (-1 * 7)) + 1 * ((2 * -4) - (1 * 7))
---STEP 4: Calculate the values inside the brackets.
= 1 * (1 - 4) + 2 * (2 - (-7)) + 1 * (-8 - 7)
---STEP 5: Simplify further.
= 1 * (-3) + 2 * (2 + 7) + 1 * (-15)
---STEP 6: Perform the multiplications.
= -3 + 2 * 9 + (-15)
= -3 + 18 - 15
---STEP 7: Add the terms.
= 15 - 15
= 0
---ANSWER: Since the scalar triple product is 0, the vectors a, b, and c are coplanar.

Why It Matters

Understanding coplanarity helps engineers design stable structures like bridges and buildings, ensuring all forces lie on the same plane for safety. In computer graphics, it's crucial for rendering realistic 3D objects and animations in video games or movie special effects. Doctors use it in medical imaging to understand how different body parts are positioned relative to each other.

Common Mistakes

MISTAKE: Confusing coplanarity with collinearity (vectors lying on the same line). | CORRECTION: Coplanarity means vectors are on the same flat surface, while collinearity means they are parallel to each other and can be on the same line. Coplanar vectors don't have to be parallel.

MISTAKE: Incorrectly calculating the determinant for the scalar triple product. | CORRECTION: Always remember the alternating signs (+ - +) when expanding the determinant along a row or column, and be careful with negative numbers in multiplication.

MISTAKE: Assuming any two vectors are always coplanar. | CORRECTION: While any two vectors always define a plane (and are thus coplanar with each other), the condition for coplanarity applies to three or more vectors. The scalar triple product specifically checks for three vectors.

Practice Questions

Try It Yourself

QUESTION: Are the vectors u = i + j, v = j + k, and w = k + i coplanar? | ANSWER: No, they are not coplanar. Their scalar triple product is 2.

QUESTION: For what value of 'm' are the vectors a = 2i - j + k, b = i + 2j - 3k, and c = 3i + mj + 5k coplanar? | ANSWER: m = -4

QUESTION: If the position vectors of four points A, B, C, D are a, b, c, d respectively, what is the condition for these four points to be coplanar? (Hint: Form three vectors from these points). | ANSWER: The vectors AB = b-a, AC = c-a, and AD = d-a must be coplanar. So, their scalar triple product (b-a) . ((c-a) x (d-a)) must be 0.

MCQ

Quick Quiz

If three vectors a, b, and c are coplanar, what is the value of their scalar triple product?

Undefined

A non-zero scalar value

The Correct Answer Is:

The scalar triple product represents the volume of the parallelepiped formed by the three vectors. If the vectors are coplanar, they lie on the same plane, meaning the 'height' of this parallelepiped is zero, hence its volume is also zero. So, the scalar triple product must be 0.

Real World Connection

In the Real World

When ISRO scientists launch rockets into space, they need to ensure the forces acting on the rocket and the trajectory it follows are carefully calculated. Understanding coplanarity helps them determine if different force vectors are aligned in a way that keeps the rocket stable and on its intended path, preventing unwanted rotations or deviations during launch and flight.

Key Vocabulary

Key Terms

VECTOR: A quantity having both magnitude and direction. | COPLANAR: Lying on the same plane. | SCALAR TRIPLE PRODUCT: A scalar quantity obtained by taking the dot product of one vector with the cross product of two other vectors. | DETERMINANT: A scalar value calculated from the elements of a square matrix, used here to find the scalar triple product. | PARALLELEPIPED: A 3D figure formed by six parallelograms.

What's Next

What to Learn Next

Great job understanding coplanarity! Next, you can explore the concept of the vector triple product, which is another way to combine three vectors. It's related to what you just learned and will help you solve even more complex problems in geometry and physics, opening doors to advanced topics!