Explanation Given: The planes x + y + z = 1 and 3 x − 2 y + z = 5 Key concepts applied: Equation of a line The vector form of the equation of a line is given by, r ( t ) = r o + t v ,Where r o is a point on the line and v is the vector parallel to the line. Equation of a plane The equation of a plane in 3D space is defined with normal vector n = 〈 a , b , c 〉 (perpendicular to the plane) and a known point p 1 = ( x 1 , y 1 , z 1 ) on the plane. n ⋅ ( p − p 1 ) = 0 a x + b y + c z − ( a x 1 + b y 1 + c z 1 ) = 0 o r a x + b y + c z + d = 0 w h e r e d = − ( a x 1 + b y 1 + c z 1 ) The intersection of two planes is a line and the intersection of three planes is a point. Vector cross product The cross product of two vectors a and b , when their magnitudes ‖ a ‖ and ‖ b ‖ and the angle between them θ are known is given by the formula, a × b = ‖ a ‖ ⋅ ‖ b ‖ ⋅ sin ( θ ) n Where n is the unit vector at right angles to both a and b The cross product of the given two vectors a = 〈 a 1 , a 2 , a 3 〉 and b = 〈 b 1 , b 2 , b 3 〉 can also be calculated using the formula below, a × b = 〈 c 1 , − c 2 , c 3 〉 where c 1 = a 2 b 3 − a 3 b 2 c 2 = a 1 b 3 − a 3 b 1 c 3 = a 1 b 2 − a 2 b 1 Considering i, j and k to be the standard unit vectors in R 3 (assuming the right handed coordinate system), we can write the cross product in the form of the determinant as follows, a × b = | i j k a i a j a k b i b j b k | The cross product of two vectors is another vector that is at right angles to both. Evaluating determinants The determinant of a square matrix A = [ a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 ] is calculated as follows, | A | = a 1 ( b 2 c 3 − b 3 c 2 ) − a 2 ( b 1 c 3 − b 3 c 1 ) + a 3 ( b 1 c 2 − b 2 c 1 ) Calculation: Let the given planes have normal vectors a = 〈 1 , 1 , 1 〉 and b = 〈 3 , − 2 , 1 〉 ...

4th Edition

ISBN: 9781319055844

Author: FRANZOSA

Publisher: VST

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Coordinate Geometry

Coordinate geometry, also known as analytic geometry or Cartesian geometry in classical mathematics, is a type of geometry that is studied using a coordinate system. Coordinate geometry is a branch of mathematics that describes the position of points on …

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System that uses coordinates to uniquely determine the position of points. The most common coordinate system is the Cartesian system, where points are given by distance along a horizontal x-axis and vertical y-axis from the origin. A polar coordinate system locates a point by its direction relative to a reference direction and its distance from a given point. In three dimensions, it leads to cylindrical and spherical coordinates.

Chapter 13, Problem 52CRE

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The intersection of the given planes

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Chapter 13, Problem 51CRE

Chapter 13, Problem 53CRE

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The intersection of the given planes

Solution Summary: The author explains how to calculate the intersection of the given planes. The equation of a line is given by r o + t v.

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To calculate: The intersection of the given planes

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Chapter 13 Solutions

To calculate:

The intersection of the given planes