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How we find a perpendicular vector to be a given vector such that ax+by+cz
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How we find a perpendicular vector to be a given vector such that ax+b...
Understanding Perpendicular Vectors
To find a vector that is perpendicular to a given vector defined by the equation ax + by + cz = 0, follow these steps:
1. Identify the Given Vector
- The coefficients (a, b, c) in the equation represent the components of the normal vector.
- The given vector is represented as V = (a, b, c).
2. Determine the Perpendicular Vector
- A vector U = (x1, y1, z1) is perpendicular to V if the dot product V • U = 0.
- This means a*x1 + b*y1 + c*z1 = 0.
3. Choose Values for Two Components
- You can select arbitrary values for two components of the perpendicular vector.
- For example, set x1 = 1 and y1 = 0. Then, solve for z1:
- Substitute into the dot product equation:
- a*1 + b*0 + c*z1 = 0
- This simplifies to z1 = -a/c (if c ≠ 0).
4. Construct the Perpendicular Vector
- Using the values chosen:
- If x1 = 1 and y1 = 0, then the perpendicular vector can be U = (1, 0, -a/c).
- Ensure to check your selected values do not result in division by zero.
5. Verify Perpendicularity
- To confirm, compute the dot product:
- V • U = a*1 + b*0 + c*(-a/c) = a - a = 0.
This confirms that the vector U is indeed perpendicular to the vector defined by ax + by + cz = 0.
To find a vector that is perpendicular to a given vector defined by the equation ax + by + cz = 0, follow these steps:
1. Identify the Given Vector
- The coefficients (a, b, c) in the equation represent the components of the normal vector.
- The given vector is represented as V = (a, b, c).
2. Determine the Perpendicular Vector
- A vector U = (x1, y1, z1) is perpendicular to V if the dot product V • U = 0.
- This means a*x1 + b*y1 + c*z1 = 0.
3. Choose Values for Two Components
- You can select arbitrary values for two components of the perpendicular vector.
- For example, set x1 = 1 and y1 = 0. Then, solve for z1:
- Substitute into the dot product equation:
- a*1 + b*0 + c*z1 = 0
- This simplifies to z1 = -a/c (if c ≠ 0).
4. Construct the Perpendicular Vector
- Using the values chosen:
- If x1 = 1 and y1 = 0, then the perpendicular vector can be U = (1, 0, -a/c).
- Ensure to check your selected values do not result in division by zero.
5. Verify Perpendicularity
- To confirm, compute the dot product:
- V • U = a*1 + b*0 + c*(-a/c) = a - a = 0.
This confirms that the vector U is indeed perpendicular to the vector defined by ax + by + cz = 0.
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How we find a perpendicular vector to be a given vector such that ax+by+cz
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How we find a perpendicular vector to be a given vector such that ax+by+cz for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about How we find a perpendicular vector to be a given vector such that ax+by+cz covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for How we find a perpendicular vector to be a given vector such that ax+by+cz.
How we find a perpendicular vector to be a given vector such that ax+by+cz for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about How we find a perpendicular vector to be a given vector such that ax+by+cz covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for How we find a perpendicular vector to be a given vector such that ax+by+cz.
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