We know that a line can be described by two points or by a single point (a, b, c) and vector v that lies on the line. A vector form for the line is given by tv + b, where b = Similarly, a plane can be described by three points or by one point and a vector perpendicular to the plane. In this case, the vector is said to be a normal vector which is perpendicular to all the vectors in the plane. Suppose Q(a, b, c) is the point in the plane and n = is a normal vector that is perpendicular to the plane. If P(x, y, z) is an arbitrary point in the same plane, then QP is a vector in the plane that is perpendicular to n. a) What is dot(QP, n)? The point-normal form of a plane is A(x - a) + B(y- b) + C(z - c) = 0. This is often written as Ax+By+ Cz= D, where D = A *a+B *b+ C * c. Consider the plane given by the equation 2x + 3y -z = 3. The vector <2, 3, -1> is normal to the plane, which is to say that <2, 3, -1> is perpendicular to every vector in the plane. b) is the point P(1, 2, 5) in the plane? c) Is the point R(-1, 3, 0) in the plane? d) Is the point S(-2, 3, 2) in the plane? e) If T(2, 2, z) is in the plane, then what is z? f) An equation for the plane parallel to the plane given by 2x + 3y -z = 3 that passes through the point (1, 1, 1) is 2x + 3y -z = Vectors also allow us to find the distance from a point to a plane, which is helpful in many ways.
We know that a line can be described by two points or by a single point (a, b, c) and vector v that lies on the line. A vector form for the line is given by tv + b, where b = Similarly, a plane can be described by three points or by one point and a vector perpendicular to the plane. In this case, the vector is said to be a normal vector which is perpendicular to all the vectors in the plane. Suppose Q(a, b, c) is the point in the plane and n = is a normal vector that is perpendicular to the plane. If P(x, y, z) is an arbitrary point in the same plane, then QP is a vector in the plane that is perpendicular to n. a) What is dot(QP, n)? The point-normal form of a plane is A(x - a) + B(y- b) + C(z - c) = 0. This is often written as Ax+By+ Cz= D, where D = A *a+B *b+ C * c. Consider the plane given by the equation 2x + 3y -z = 3. The vector <2, 3, -1> is normal to the plane, which is to say that <2, 3, -1> is perpendicular to every vector in the plane. b) is the point P(1, 2, 5) in the plane? c) Is the point R(-1, 3, 0) in the plane? d) Is the point S(-2, 3, 2) in the plane? e) If T(2, 2, z) is in the plane, then what is z? f) An equation for the plane parallel to the plane given by 2x + 3y -z = 3 that passes through the point (1, 1, 1) is 2x + 3y -z = Vectors also allow us to find the distance from a point to a plane, which is helpful in many ways.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:We know that a line can be described by two points or by a single point (a, b, c) and vector v that lies on the line. A vector form for the line is given by tv + b, where b = <a, b, c>.
Similarly, a plane can be described by three points or by one point and a vector perpendicular to the plane. In this case, the vector is said to be a normal vector which is perpendicular to all the vectors in the plane.
Suppose Q(a, b, c) is the point in the plane and n = <A, B, C> is a normal vector that is perpendicular to the plane. If P(x, y, z) is an arbitrary point in the same plane, then
QP = <x-a, y - b, z - c>
is a vector in the plane that is perpendicular to n.
a) What is dot(QP, n)?
The point-normal form of a plane is A(x - a) + B(y - b) + C(z - c) = 0. This is often written as Ax+By+ Cz = D, where D = A *a+B*b+ C*c.
Consider the plane given by the equation 2x + 3y -z = 3. The vector <2, 3, -1> is normal to the plane, which is to say that <2, 3, -1> is perpendicular to every vector in the plane.
b) is the point P(1, 2, 5) in the plane?
c) Is the point R(-1, 3, 0) in the plane?
d) Is the point S(-2, 3, 2) in the plane?
e) If T(2, 2, z) is in the plane, then what is z?
f) An equation for the plane parallel to the plane given by 2x + 3y - z = 3 that passes through the point (1, 1, 1) is 2x + 3y -z =
Vectors also allow us to find the distance from a point to a plane, which is helpful in many ways.
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