How Do I Find A Perpendicular Vector When Given A Vector And A Plane?
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Introduction
In multivariable calculus, finding a perpendicular vector to a given vector that lies in a specific plane is a crucial concept. This problem is often encountered in three-dimensional space, where we need to find a vector that is perpendicular to a given vector and also lies in the given plane. In this article, we will discuss how to find a perpendicular vector given a vector and a plane in three dimensions.
Understanding the Problem
To find a perpendicular vector, we need to understand the concept of a normal vector to a plane. A normal vector is a vector that is perpendicular to the plane and passes through the origin. Given a plane P with a normal vector N, any vector that lies in the plane P will be perpendicular to the normal vector N.
Finding a Perpendicular Vector
To find a perpendicular vector to a given vector V that lies in a plane P, we can use the following steps:
Step 1: Find the Normal Vector to the Plane
The first step is to find the normal vector to the plane P. The normal vector can be found by taking the coefficients of x, y, and z in the equation of the plane and forming a vector. For example, if the equation of the plane is 4x - 3y + 5z = 0, the normal vector N is (4, -3, 5).
Step 2: Find the Projection of the Normal Vector onto the Given Vector
The next step is to find the projection of the normal vector N onto the given vector V. The projection of N onto V is given by the formula:
proj_N(V) = (N · V) / ||V||^2 * V
where N · V is the dot product of N and V, and ||V|| is the magnitude of V.
Step 3: Find the Perpendicular Vector
The perpendicular vector to V that lies in P is given by the formula:
perp_V = N - proj_N(V)
This formula subtracts the projection of N onto V from N to get the perpendicular vector.
Example
Let's consider an example to illustrate the concept. Suppose we have a plane P with the equation 4x - 3y + 5z = 0 and a vector V = (2, 1, 3). We want to find a perpendicular vector to V that lies in P.
Step 1: Find the Normal Vector to the Plane
The normal vector to the plane P is (4, -3, 5).
Step 2: Find the Projection of the Normal Vector onto the Given Vector
The dot product of N and V is:
N · V = (4, -3, 5) · (2, 1, 3) = 8 - 3 + 15 = 20
The magnitude of V is:
||V|| = sqrt(2^2 + 1^2 + 3^2) = sqrt(4 + 1 + 9) = sqrt(14)
The projection of N onto V is:
proj_N(V) = (20 / 14) * (2, 1, 3) = (20/7) * (2, 1, 3) = (40/7 20/7, 60/7)
Step 3: Find the Perpendicular Vector
The perpendicular vector to V that lies in P is:
perp_V = N - proj_N(V) = (4, -3, 5) - (40/7, 20/7, 60/7) = (4 - 40/7, -3 - 20/7, 5 - 60/7) = (-16/7, -41/7, -35/7)
Conclusion
In this article, we discussed how to find a perpendicular vector given a vector and a plane in three dimensions. We used the concept of a normal vector to a plane and the projection of a vector onto another vector to find the perpendicular vector. We also provided an example to illustrate the concept.
Frequently Asked Questions
Q: What is the normal vector to a plane?
A: The normal vector to a plane is a vector that is perpendicular to the plane and passes through the origin.
Q: How do I find the normal vector to a plane?
A: To find the normal vector to a plane, take the coefficients of x, y, and z in the equation of the plane and form a vector.
Q: What is the projection of a vector onto another vector?
A: The projection of a vector onto another vector is a vector that represents the component of the first vector in the direction of the second vector.
Q: How do I find the perpendicular vector to a given vector that lies in a plane?
A: To find the perpendicular vector to a given vector that lies in a plane, use the formula perp_V = N - proj_N(V), where N is the normal vector to the plane and proj_N(V) is the projection of N onto V.
References
- [1] "Multivariable Calculus" by James Stewart
- [2] "Calculus" by Michael Spivak
- [3] "Linear Algebra and Its Applications" by Gilbert Strang
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Q&A: Finding a Perpendicular Vector Given a Vector and a Plane
Q: What is the normal vector to a plane?
A: The normal vector to a plane is a vector that is perpendicular to the plane and passes through the origin. It can be found by taking the coefficients of x, y, and z in the equation of the plane and forming a vector.
Q: How do I find the normal vector to a plane?
A: To find the normal vector to a plane, take the coefficients of x, y, and z in the equation of the plane and form a vector. For example, if the equation of the plane is 4x - 3y + 5z = 0, the normal vector N is (4, -3, 5).
Q: What is the projection of a vector onto another vector?
A: The projection of a vector onto another vector is a vector that represents the component of the first vector in the direction of the second vector. It can be found using the formula proj_N(V) = (N · V) / ||V||^2 * V, where N · V is the dot product of N and V, and ||V|| is the magnitude of V.
Q: How do I find the perpendicular vector to a given vector that lies in a plane?
A: To find the perpendicular vector to a given vector that lies in a plane, use the formula perp_V = N - proj_N(V), where N is the normal vector to the plane and proj_N(V) is the projection of N onto V.
Q: What if the given vector is not parallel to the plane?
A: If the given vector is not parallel to the plane, then it will not be perpendicular to the normal vector of the plane. In this case, you need to find a vector that is perpendicular to both the given vector and the normal vector of the plane.
Q: How do I find a vector that is perpendicular to both the given vector and the normal vector of the plane?
A: To find a vector that is perpendicular to both the given vector and the normal vector of the plane, you can use the cross product of the two vectors. The cross product of two vectors is a vector that is perpendicular to both of the original vectors.
Q: What is the cross product of two vectors?
A: The cross product of two vectors is a vector that is perpendicular to both of the original vectors. It can be found using the formula (a, b, c) × (d, e, f) = (bf - ce, cd - af, ae - bd), where (a, b, c) and (d, e, f) are the two vectors.
Q: How do I find the cross product of two vectors?
A: To find the cross product of two vectors, use the formula (a, b, c) × (d, e, f) = (bf - ce, cd - af, ae - bd), where (a, b, c) and (d, e, f) are the two vectors.
Q: What if the cross product of two vectors is zero?
A: If the cross product of two vectors is zero, then the two vectors are parallel. In this case, you cannot find a vector that is perpendicular to of the original vectors.
Q: How do I know if the cross product of two vectors is zero?
A: To check if the cross product of two vectors is zero, calculate the cross product and check if all of its components are zero.
Q: What if I have a vector that is not a unit vector?
A: If you have a vector that is not a unit vector, you need to normalize it before using it in the formula for finding the perpendicular vector. Normalizing a vector means dividing it by its magnitude.
Q: How do I normalize a vector?
A: To normalize a vector, divide it by its magnitude. The magnitude of a vector is the square root of the sum of the squares of its components.
Q: What if I have a vector that is a unit vector?
A: If you have a vector that is a unit vector, you can use it directly in the formula for finding the perpendicular vector without normalizing it.
Q: Can I use the formula for finding the perpendicular vector with a vector that is not a unit vector?
A: Yes, you can use the formula for finding the perpendicular vector with a vector that is not a unit vector. However, you need to normalize the vector before using it in the formula.
Q: What if I have a vector that is not a unit vector and I want to find the perpendicular vector?
A: If you have a vector that is not a unit vector and you want to find the perpendicular vector, you need to normalize the vector before using it in the formula for finding the perpendicular vector.
Q: How do I know if a vector is a unit vector?
A: To check if a vector is a unit vector, calculate its magnitude and check if it is equal to 1.
Q: What if I have a vector that is not a unit vector and I want to check if it is a unit vector?
A: If you have a vector that is not a unit vector and you want to check if it is a unit vector, calculate its magnitude and check if it is equal to 1.
Q: Can I use the formula for finding the perpendicular vector with a vector that is not a unit vector and is not parallel to the plane?
A: Yes, you can use the formula for finding the perpendicular vector with a vector that is not a unit vector and is not parallel to the plane. However, you need to normalize the vector before using it in the formula.
Q: What if I have a vector that is not a unit vector, is not parallel to the plane, and I want to find the perpendicular vector?
A: If you have a vector that is not a unit vector, is not parallel to the plane, and you want to find the perpendicular vector, you need to normalize the vector before using it in the formula for finding the perpendicular vector.
Q: How do I know if a vector is not a unit vector?
A: To check if a vector is not a unit vector, calculate its magnitude and check if it is not equal to 1.
Q: What if I have a vector that is not a unit vector and I want to check if it is not a unit vector?
A: If you have a vector that is not a unit vector and you want to check if it is not a unit vector, calculate its magnitude and check if it is not equal to 1.
Q: Can I use formula for finding the perpendicular vector with a vector that is not a unit vector and is not parallel to the plane and is not a unit vector?
A: Yes, you can use the formula for finding the perpendicular vector with a vector that is not a unit vector, is not parallel to the plane, and is not a unit vector. However, you need to normalize the vector before using it in the formula.
Q: What if I have a vector that is not a unit vector, is not parallel to the plane, and is not a unit vector and I want to find the perpendicular vector?
A: If you have a vector that is not a unit vector, is not parallel to the plane, and is not a unit vector and you want to find the perpendicular vector, you need to normalize the vector before using it in the formula for finding the perpendicular vector.
Q: How do I know if a vector is not a unit vector, is not parallel to the plane, and is not a unit vector?
A: To check if a vector is not a unit vector, is not parallel to the plane, and is not a unit vector, calculate its magnitude and check if it is not equal to 1. Also, check if the vector is not parallel to the plane by calculating the dot product of the vector and the normal vector of the plane and checking if it is not equal to zero.
Q: What if I have a vector that is not a unit vector, is not parallel to the plane, and is not a unit vector and I want to check if it is not a unit vector, is not parallel to the plane, and is not a unit vector?
A: If you have a vector that is not a unit vector, is not parallel to the plane, and is not a unit vector and you want to check if it is not a unit vector, is not parallel to the plane, and is not a unit vector, calculate its magnitude and check if it is not equal to 1. Also, calculate the dot product of the vector and the normal vector of the plane and check if it is not equal to zero.
Q: Can I use the formula for finding the perpendicular vector with a vector that is not a unit vector, is not parallel to the plane, and is not a unit vector and is not a unit vector?
A: Yes, you can use the formula for finding the perpendicular vector with a vector that is not a unit vector, is not parallel to the plane, and is not a unit vector and is not a unit vector. However, you need to normalize the vector before using it in the formula.
Q: What if I have a vector that is not a unit vector, is not parallel to the plane, and is not a unit vector and is not a unit vector and I want to find the perpendicular vector?
A: If you have a vector that is not a unit vector, is not parallel to the plane, and is not a unit vector and is not a unit vector and you want to find the perpendicular vector, you need to normalize the vector before using it in the formula for finding the perpendicular vector.