Explanation Given information: Given, the line with vector parametrization r ( t ) = 〈 3 , 1 , − 4 〉 + t 〈 − 2 , − 2 , 3 〉 intersects the sphere ( x − 1 ) 2 + ( y + 3 ) 2 + z 2 = 8 in two points. Calculation: A point on the line is of the form ( 3 − 2 t , 1 − 2 t , − 4 + 3 t ) When the line intersects the sphere, the point needs to satisfy the equation of the sphere. Hence plugging the point in the equation of sphere to solve for t : ( x − 1 ) 2 + ( y + 3 ) 2 + z 2 = 8 ( 3 − 2 t − 1 ) 2 + ( 1 − 2 t + 3 ) 2 + ( − 4 + 3 t ) 2 = 8 ( 2 − 2 t ) 2 + ( 4 − 2 t ) 2 + ( − 4 + 3 t ) 2 = 8 ( 4 − 8 t + 4 t 2 ) + ( 16 − 16 t + 4 t 2 ) + ( 16 − 24 t + 9 t 2 ) = 8 36 − 48 t + 17 t 2 = 8 17 t 2 − 48 t + 28 = 0 Solving the quadratic equation for t : t = 2 , 14 17 When t = 2 , the poi

4th Edition

ISBN: 9781319055905

Author: FRANZOSA

Publisher: VST

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Vectors are those objects which have a magnitude along with the direction. In vector arithmetic, we will see how arithmetic operators like addition and multiplication are used on any two vectors. Arithmetic in basic means dealing with numbers. Here, magn…

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Vector calculus is an important branch of mathematics and it relates two important branches of mathematics namely vector and calculus.

Question

Chapter 12.2, Problem 67E

To determine

To find the points of intersections of the given line and sphere.

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Chapter 12.2, Problem 66E

Chapter 12.2, Problem 68E

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Chapter 12 Solutions

EBK CALCULUS: EARLY TRANSCENDENTALS

Ch. 12.1 - Prob. 1PQ Ch. 12.1 - Prob. 2PQ Ch. 12.1 - Prob. 3PQ Ch. 12.1 - Prob. 4PQ Ch. 12.1 - Prob. 5PQ Ch. 12.1 - Prob. 6PQ Ch. 12.1 - Prob. 1E Ch. 12.1 - Prob. 2E Ch. 12.1 - Prob. 3E Ch. 12.1 - Prob. 4E

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To find the points of intersections of the given line and sphere.

Solution Summary: The author explains how to find the intersections of the given line and sphere.

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To find the points of intersections of the given line and sphere.

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Chapter 12 Solutions