UNF MAC 2313 - Lines and Planes (2 pages)

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Lines and Planes



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Lines and Planes

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School:
University of North Florida
Course:
Mac 2313 - (gm) Calculus III
(gm) Calculus III Documents
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MAC 2313 LINES AND PLANES Exercise 1 You are given vectors a h1 2 2i and b h2 0 3i For what value of t does the vector a t b lie in the plane determined by the vectors p h1 1 1i and q h1 2 3i Hint Try to use triple scalar product The final answer is t 7 Exercise 2 Use properties of the cross product and triple scalar product to simplify the following expressions i a b b a ii a b a Exercise 3 You are given the points A 1 5 1 B 3 2 2 and C 1 3 a where a is a real number Let l be the line determined by A and the vector s h1 1 3i Find the value of a so that the plane determined by A B and C contains the line l Answer a 15 Exercise 4 Consider the line x 4 y 3 z 3 l 3 4 2 Find an equation of the line that passes through the point 3 2 0 and is perpendicular to l Hint First find an equation of the plane containing the point 3 2 0 and is perpendicular to l Use this plane to get the second point on the line perpendicular to l The final answer is x 3 y 2 z 0 4 9 5 Exercise 5 Let l1 be defined as the line of intersection of the planes y 3 and 2x y z 6 Additionally let the line l2 be defined as x 1 y 1 z 2 l2 2 2 1 Show that the lines l1 and l2 belong to the same plane Hint You need to show that l1 and l2 intersect at some point or that l1 and l2 are parallel In other words you need to show that l1 and l2 are not skew lines 1 2 Exercise 6 You are given the planes 1 2x y z 7 2 x y 4 Let m be the line defined as the intersection of the planes 1 and 2 You are also given the line x 2 y 1 z 1 l 0 1 4 Let T1 and T2 be the points of intersection of l with the planes 1 and 2 respectively Find the projections of T1 and T2 onto m Hint First you need to find T1 and T2 You should get T1 2 0 3 and T2 2 2 5 Then you need to find the line m and you will get m x 4 y 0 z 1 The projection of T1 onto m is the intersection 1 1 1 of m and the plane through T1 perpendicular to m The same holds for T2 The final answer is as follows the projections of T1 and T2 onto m are 2 2 1 and 4 0 1 respectively Exercise 7 Consider the plane x 3y 2z 5 0 and the sphere of radius 12 centered at 5 14 9 i Find the point P on with the smallest distance from the sphere ii How far is P from the center of the sphere iii How far is P from the sphere Answer i P 0 1 1 ii 5 14 iii 5 14 12 Exercise 8 Let S be the sphere with the following properties the radius of S is 3 the sphere S touches the plane 6x 3y 6z 9 at the point 1 1 1 and S is located in the half space 6x 3y 6z 9 i Write an equation of the sphere S ii Find all intersection points of the sphere S with the plane y 3 Answer i x 3 2 y 2 z 3 2 9 ii point 3 3 3


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