What is Cartesian Equation of a Plane? | Simple Explanation for Class 12

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What is the Cartesian Equation of a Plane Passing Through a Point and Perpendicular to a Vector?

Grade Level:

Class 12

AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics

Definition

What is it?

The Cartesian equation of a plane is a mathematical rule (like a formula) that describes all the points lying on a flat surface in 3D space. It helps us define a plane when we know one point it passes through and a vector (a line with direction) that is exactly perpendicular to it, like a wall standing straight up from the floor.

Simple Example

Quick Example

Imagine your school blackboard is a plane. If you know the exact coordinates of one corner of the blackboard (say, where the chalk dust collects) and you know the direction of the wall it's on (which is perpendicular to the blackboard's surface), you can write an equation that describes every single point on that blackboard.

Worked Example

Step-by-Step

Let's find the Cartesian equation of a plane passing through point A(1, 2, 3) and perpendicular to the vector N = 2i + 3j - 4k.

Step 1: Identify the given point (x1, y1, z1) and the normal vector's components (a, b, c).
Here, (x1, y1, z1) = (1, 2, 3) and (a, b, c) = (2, 3, -4).

---Step 2: Recall the formula for the Cartesian equation of a plane: a(x - x1) + b(y - y1) + c(z - z1) = 0.

---Step 3: Substitute the values into the formula.
2(x - 1) + 3(y - 2) + (-4)(z - 3) = 0

---Step 4: Distribute the numbers.
2x - 2 + 3y - 6 - 4z + 12 = 0

---Step 5: Combine the constant terms.
2x + 3y - 4z + (-2 - 6 + 12) = 0
2x + 3y - 4z + 4 = 0

The Cartesian equation of the plane is 2x + 3y - 4z + 4 = 0.

Why It Matters

Understanding plane equations is super important in fields like AI/ML for creating 3D models and in Engineering for designing car parts or buildings. Doctors use it in Medicine to map out organs from MRI scans, and even in Space Technology to plot satellite orbits around Earth.

Common Mistakes

MISTAKE: Forgetting to put the minus sign with the point coordinates in the formula, like writing a(x + x1) instead of a(x - x1). | CORRECTION: Always remember the formula is a(x - x1) + b(y - y1) + c(z - z1) = 0, so subtract the point's coordinates.

MISTAKE: Mixing up the components of the normal vector with the coordinates of the point. | CORRECTION: The coefficients 'a, b, c' come from the normal vector, while 'x1, y1, z1' come from the point the plane passes through.

MISTAKE: Making calculation errors when distributing or combining constant terms, especially with negative numbers. | CORRECTION: Double-check your arithmetic carefully, especially when multiplying or adding negative numbers. For example, -4 * -3 is +12, not -12.

Practice Questions

Try It Yourself

QUESTION: Find the Cartesian equation of a plane passing through the point (2, -1, 4) and perpendicular to the vector 3i - 2j + 5k. | ANSWER: 3x - 2y + 5z - 28 = 0

QUESTION: A plane passes through the origin (0, 0, 0) and is perpendicular to the vector i + j + k. What is its Cartesian equation? | ANSWER: x + y + z = 0

QUESTION: A plane passes through the point P(5, 0, -2) and is perpendicular to the line joining the points Q(1, 2, 3) and R(3, -1, 4). Find the Cartesian equation of the plane. (Hint: First find the vector QR, which will be the normal vector). | ANSWER: 2x - 3y + z - 8 = 0

MCQ

Quick Quiz

Which of the following is the Cartesian equation of a plane passing through (0, 1, 0) and perpendicular to the vector i - 2j + 3k?

x + 2y + 3z - 2 = 0

x - 2y + 3z + 2 = 0

x - 2y + 3z - 2 = 0

x + 2y + 3z + 2 = 0

The Correct Answer Is:

Using a(x-x1) + b(y-y1) + c(z-z1) = 0, with (x1,y1,z1)=(0,1,0) and (a,b,c)=(1,-2,3), we get 1(x-0) - 2(y-1) + 3(z-0) = 0, which simplifies to x - 2y + 2 + 3z = 0, or x - 2y + 3z + 2 = 0.

Real World Connection

In the Real World

Imagine you're designing a new mobile phone. The screen's flat surface can be represented by a plane equation. Engineers use these equations to precisely define the angles and positions of different components inside the phone, ensuring everything fits perfectly and works. This is crucial in fields like CAD (Computer-Aided Design) software used in companies like Tata Motors or for designing intricate circuit boards.

Key Vocabulary

Key Terms

CARTESIAN EQUATION: An equation that uses x, y, z coordinates to describe a geometric object like a plane. | PLANE: A flat, two-dimensional surface that extends infinitely in 3D space. | PERPENDICULAR: At a right angle (90 degrees) to another line or surface. | NORMAL VECTOR: A vector that is perpendicular to a surface (like a plane).

What's Next

What to Learn Next

Great job learning about plane equations! Next, you can explore the 'Vector Equation of a Plane' and 'Equation of a Plane Passing Through Three Non-Collinear Points'. These concepts build directly on what you've learned and will help you tackle more complex 3D geometry problems.