Explanation Given: The set of all points X = ( x , y , z ) equidistant from two points P , Q in R 3 is a plane. The figure is: Calculation: The set of all points X = ( x , y , z ) equidistant from two points P , Q in R 3 is a plane. Let the point P = ( a , b , c ) and Q = ( p , q , r ) . Let R be the midpoint of P Q . Then, R = ( a + p 2 , b + q 2 , c + r 2 ) . The point X will be equidistant from the points P and Q if and only if X R → is orthogonal to P Q → . X R → ⋅ P Q → = 0 [ ( a + p 2 − x ) + ( b + q 2 − y ) + ( c + r 2 − z ) ] ⋅ ( p − a , q − b , r − c ) = 0 Simplify the above equation...

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 13.3, Problem 97E

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Show that $X$ lies on this plane if $\vec{P Q} \cdot \vec{O X} = \frac{1}{2} ({‖ \vec{O Q} ‖}^{2} - {‖ \vec{O P} ‖}^{2})$ .

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Chapter 13.3, Problem 96E

Chapter 13.3, Problem 98E

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Chapter 13 Solutions

Calculus - Standalone book

Ch. 13.1 - Prob. 1PQ Ch. 13.1 - Prob. 2PQ Ch. 13.1 - Prob. 3PQ Ch. 13.1 - Prob. 4PQ Ch. 13.1 - Prob. 5PQ Ch. 13.1 - Prob. 6PQ Ch. 13.1 - Prob. 1E Ch. 13.1 - Prob. 2E Ch. 13.1 - Prob. 3E Ch. 13.1 - Prob. 4E

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Show that X lies on this plane if P Q → ⋅ O X → = 1 2 ( ‖ O Q → ‖ 2 − ‖ O P → ‖ 2 ) .

Solution Summary: The author explains that the point X lies on the plane if P Q x y z is equidistant from two points P and Q in R 3.

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Show that $X$ lies on this plane if $\vec{P Q} \cdot \vec{O X} = \frac{1}{2} ({‖ \vec{O Q} ‖}^{2} - {‖ \vec{O P} ‖}^{2})$ .

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Chapter 13 Solutions

Show that X lies on this plane if P Q → ⋅ O X → = 1 2 ( ‖ O Q → ‖ 2 − ‖ O P → ‖ 2 ) .

Solution Summary: The author explains that the point X lies on the plane if P Q x y z is equidistant from two points P and Q in R 3.

Concept explainers

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Show that X lies on this plane if PQ→⋅OX→=12(‖OQ→‖2−‖OP→‖2).

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Chapter 13 Solutions

Show that $X$ lies on this plane if $\vec{P Q} \cdot \vec{O X} = \frac{1}{2} ({‖ \vec{O Q} ‖}^{2} - {‖ \vec{O P} ‖}^{2})$ .