Inaugurated by IN-SPACe
ISRO Registered Space Tutor
S7-SA2-0277
What is the Vector 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 vector equation of a plane passing through a specific point and perpendicular to a given vector helps us describe the plane's position and orientation in 3D space. It uses a position vector for the point and a normal vector that tells us the plane's 'tilt'.
Simple Example
Quick Example
Imagine you have a flat cutting board (the plane) and you want to place it perfectly flat on a table (passing through a point). You also want it to be perfectly straight, maybe perpendicular to the table leg (the vector). This equation helps you write down exactly how to place that cutting board using math.
Worked Example
Step-by-Step
Let's find the vector equation of a plane passing through the point A (1, 2, 3) and perpendicular to the vector n = 2i + 3j - 4k.
Step 1: Identify the position vector of the given point. The position vector 'a' for point A(1, 2, 3) is a = i + 2j + 3k.
---
Step 2: Identify the normal vector 'n' to the plane. The given normal vector is n = 2i + 3j - 4k.
---
Step 3: Recall the general vector equation of a plane passing through a point 'a' and perpendicular to 'n', which is (r - a) . n = 0.
---
Step 4: Substitute the values of 'a' and 'n' into the equation. So, (r - (i + 2j + 3k)) . (2i + 3j - 4k) = 0.
---
Step 5: This is the required vector equation. You can also expand it if needed for the Cartesian form, but this is the vector form.
Answer: The vector equation is (r - (i + 2j + 3k)) . (2i + 3j - 4k) = 0.
Why It Matters
Understanding plane equations is super important in fields like AI/ML for creating decision boundaries, in Physics for studying forces and fields, and in Engineering for designing structures. Engineers use this to ensure bridges are stable and architects use it to design buildings safely.
Common Mistakes
MISTAKE: Students often confuse the position vector of the point on the plane with the normal vector. | CORRECTION: The position vector 'a' points FROM the origin TO a point ON the plane, while the normal vector 'n' is PERPENDICULAR to the plane itself.
MISTAKE: Forgetting the dot product in the equation (r - a) . n = 0 and writing it as (r - a)n = 0 or (r - a) x n = 0. | CORRECTION: The normal vector 'n' is perpendicular to ANY vector lying IN the plane. So, the vector (r - a), which lies in the plane, must have a dot product of zero with 'n'.
MISTAKE: Incorrectly performing vector subtraction or dot product with 'r' when expanding to Cartesian form. | CORRECTION: Remember r = xi + yj + zk. So, (r - a) becomes (x-x1)i + (y-y1)j + (z-z1)k, and then perform the dot product with n = Ai + Bj + Ck as A(x-x1) + B(y-y1) + C(z-z1) = 0.
Practice Questions
Try It Yourself
QUESTION: Find the vector equation of a plane passing through the point (5, -1, 2) and perpendicular to the vector 3i - 2j + k. | ANSWER: (r - (5i - j + 2k)) . (3i - 2j + k) = 0
QUESTION: A plane passes through the origin (0,0,0) and is perpendicular to the vector 4i + 5j - 6k. Write its vector equation. | ANSWER: r . (4i + 5j - 6k) = 0
QUESTION: If the vector equation of a plane is (r - (2i + 0j - k)) . (i + j + 2k) = 0, identify the point it passes through and a normal vector to it. | ANSWER: Point: (2, 0, -1); Normal vector: i + j + 2k
MCQ
Quick Quiz
What is the vector equation of a plane passing through the point (P) and perpendicular to the vector (N)?
(r + P) . N = 0
(r - P) . N = 0
r . (P - N) = 0
r x N = P
The Correct Answer Is:
B
The correct form for a plane passing through point P (with position vector P) and perpendicular to normal vector N is (r - P) . N = 0. This is because (r - P) is a vector lying in the plane, and it must be perpendicular to the normal vector N, meaning their dot product is zero.
Real World Connection
In the Real World
Think about how self-driving cars like those being tested in Bengaluru or Pune navigate. They use sensors to map their surroundings, representing roads and obstacles as planes and points in 3D space. This math helps them understand the orientation of a road surface or the side of another vehicle, ensuring a safe ride.
Key Vocabulary
Key Terms
PLANE: A flat, two-dimensional surface that extends infinitely in 3D space. | POSITION VECTOR: A vector from the origin to a specific point. | NORMAL VECTOR: A vector that is perpendicular to a surface or a plane. | DOT PRODUCT: A scalar product of two vectors, resulting in a single number, useful for checking perpendicularity. | PERPENDICULAR: At an angle of 90 degrees to something.
What's Next
What to Learn Next
Great job learning about the vector equation of a plane! Next, you can explore the Cartesian equation of a plane. This builds on what you've learned here and shows how to write the same plane equation using x, y, and z coordinates, which is super useful for solving problems.
