Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 12: Vectors and the Geometry of Space - Questions to Guide Your Review - Page 733: 15

Answer

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Work Step by Step

To find the intersection of two lines in space, you need to determine if they intersect at a single point, are parallel and do not intersect, or are coincident (meaning they lie on top of each other). 1. Intersection of Two Lines in Space: - If the lines intersect at a single point, you can find the coordinates of the intersection point by solving the system of equations formed by the parametric equations of the two lines. - For example, consider the following lines: Line 1: $x = 2 + t$, $y = 3 - t$ ,$ z = 1 + 2t$ Line 2: $x = 4 - 2s$ , $y = 1 + 3s$ , $z = 5 - s$ By equating the corresponding components of the two lines, you can solve for the values of t and s, which will give you the coordinates of the intersection point. 2. Intersection of a Line and a Plane: - To find the intersection of a line and a plane, you need to determine if the line lies on the plane, is parallel to the plane and does not intersect, or intersects the plane at a single point. - For example, consider the following line and plane: Line: $x = 2 + t$, $y = 3 - t$ , $ z = 1 + 2t$ Plane: $2x + 3y - z = 7$ By substituting the parametric equations of the line into the equation of the plane, you can solve for the value of t. If a valid value of t is obtained, it means the line intersects the plane at a single point. 3. Intersection of Two Planes: - To find the intersection of two planes, you need to determine if they intersect along a line, are parallel and do not intersect, or coincide (meaning they are the same plane). - For example, consider the following planes: Plane 1: $2x + 3y - z = 7$ Plane 2: $x - 2y + 4z = 5$ By setting the equations of the two planes equal to each other, you can solve for the values of $x$, $y$, and $z$. If a valid solution is obtained, it means the planes intersect along a line. It's important to note that in some cases, the lines or planes may not intersect or coincide, resulting in no solution.
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