JEE Exam > JEE Questions > If 𝐴(−1,2,3),𝐵(1,1,1) and 𝐶(2, −1,3) are p...
Start Learning for Free
If 𝐴(−1,2,3),𝐵(1,1,1) and 𝐶(2, −1,3) are points on a plane. A unit normal vector to the plane
ABC is?
ABC is?
Most Upvoted Answer
If 𝐴(−1,2,3),𝐵(1,1,1) and 𝐶(2, −1,3) are points on a plane. A unit ...
Points on the Plane ABC
The given points A(-1,2,3), B(1,1,1), and C(2,-1,3) are points on the plane ABC.
Equation of a Plane
The equation of a plane in three-dimensional space can be written in the form Ax + By + Cz + D = 0, where A, B, C are the coefficients of x, y, z, respectively, and D is a constant.
Finding the Equation of the Plane ABC
To find the equation of the plane ABC, we need to find the coefficients A, B, C, and D.
1. Finding the Normal Vector
The normal vector of a plane is perpendicular to the plane. To find the normal vector, we can find the cross product of two vectors lying on the plane.
Let's take two vectors AB and AC lying on the plane ABC.
AB = B - A = (1, 1, 1) - (-1, 2, 3) = (2, -1, -2)
AC = C - A = (2, -1, 3) - (-1, 2, 3) = (3, -3, 0)
Now, we can find the cross product of AB and AC to get the normal vector.
AB x AC = (2, -1, -2) x (3, -3, 0) = (6, -6, -9)
2. Normalizing the Normal Vector
To get a unit normal vector, we need to normalize the normal vector by dividing it by its magnitude.
Magnitude of the normal vector = √(6^2 + (-6)^2 + (-9)^2) = √(36 + 36 + 81) = √153
Unit normal vector = (6/√153, -6/√153, -9/√153)
Thus, a unit normal vector to the plane ABC is (6/√153, -6/√153, -9/√153).
The given points A(-1,2,3), B(1,1,1), and C(2,-1,3) are points on the plane ABC.
Equation of a Plane
The equation of a plane in three-dimensional space can be written in the form Ax + By + Cz + D = 0, where A, B, C are the coefficients of x, y, z, respectively, and D is a constant.
Finding the Equation of the Plane ABC
To find the equation of the plane ABC, we need to find the coefficients A, B, C, and D.
1. Finding the Normal Vector
The normal vector of a plane is perpendicular to the plane. To find the normal vector, we can find the cross product of two vectors lying on the plane.
Let's take two vectors AB and AC lying on the plane ABC.
AB = B - A = (1, 1, 1) - (-1, 2, 3) = (2, -1, -2)
AC = C - A = (2, -1, 3) - (-1, 2, 3) = (3, -3, 0)
Now, we can find the cross product of AB and AC to get the normal vector.
AB x AC = (2, -1, -2) x (3, -3, 0) = (6, -6, -9)
2. Normalizing the Normal Vector
To get a unit normal vector, we need to normalize the normal vector by dividing it by its magnitude.
Magnitude of the normal vector = √(6^2 + (-6)^2 + (-9)^2) = √(36 + 36 + 81) = √153
Unit normal vector = (6/√153, -6/√153, -9/√153)
Thus, a unit normal vector to the plane ABC is (6/√153, -6/√153, -9/√153).
|
Explore Courses for JEE exam
|
|
Similar JEE Doubts
Top Courses for JEEView all
If 𝐴(−1,2,3),𝐵(1,1,1) and 𝐶(2, −1,3) are points on a plane. A unit normal vector to the plane ABC is?
Question Description
If 𝐴(−1,2,3),𝐵(1,1,1) and 𝐶(2, −1,3) are points on a plane. A unit normal vector to the plane ABC is? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If 𝐴(−1,2,3),𝐵(1,1,1) and 𝐶(2, −1,3) are points on a plane. A unit normal vector to the plane ABC is? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If 𝐴(−1,2,3),𝐵(1,1,1) and 𝐶(2, −1,3) are points on a plane. A unit normal vector to the plane ABC is?.
If 𝐴(−1,2,3),𝐵(1,1,1) and 𝐶(2, −1,3) are points on a plane. A unit normal vector to the plane ABC is? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If 𝐴(−1,2,3),𝐵(1,1,1) and 𝐶(2, −1,3) are points on a plane. A unit normal vector to the plane ABC is? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If 𝐴(−1,2,3),𝐵(1,1,1) and 𝐶(2, −1,3) are points on a plane. A unit normal vector to the plane ABC is?.
Solutions for If 𝐴(−1,2,3),𝐵(1,1,1) and 𝐶(2, −1,3) are points on a plane. A unit normal vector to the plane ABC is? in English & in Hindi are available as part of our courses for JEE.
Download more important topics, notes, lectures and mock test series for JEE Exam by signing up for free.
Here you can find the meaning of If 𝐴(−1,2,3),𝐵(1,1,1) and 𝐶(2, −1,3) are points on a plane. A unit normal vector to the plane ABC is? defined & explained in the simplest way possible. Besides giving the explanation of
If 𝐴(−1,2,3),𝐵(1,1,1) and 𝐶(2, −1,3) are points on a plane. A unit normal vector to the plane ABC is?, a detailed solution for If 𝐴(−1,2,3),𝐵(1,1,1) and 𝐶(2, −1,3) are points on a plane. A unit normal vector to the plane ABC is? has been provided alongside types of If 𝐴(−1,2,3),𝐵(1,1,1) and 𝐶(2, −1,3) are points on a plane. A unit normal vector to the plane ABC is? theory, EduRev gives you an
ample number of questions to practice If 𝐴(−1,2,3),𝐵(1,1,1) and 𝐶(2, −1,3) are points on a plane. A unit normal vector to the plane ABC is? tests, examples and also practice JEE tests.
|
|
Explore Courses for JEE exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup