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Planes in three dimensions 1.Normal equation 2.Cartesian equation of plane 3.Angle between a line and a plane or between two planes 4.Determine whether a line intersects or lies in or parallel to a plane 5.Distance from a point to a plane

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Normal equation The normal is perpendicular to any line in the plane n o A R (p,q,r) (a,b,c) (x,y,z) Cartesian equation of a plane

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theorems 1.The graph of every linear equation is a plane with normal vector (a,b,c) 2. Two planes with normal vectors a & b are (i) parallel if a and b are parallel (ii) orthogonal if a and b are orthogonal

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Extra :Cross product Definition: where is right-handedright-handed determinant where x,y,and z are unit vectors, here u x v is always perpendicular to u and v Therefore we can find the normal of the plane from two vectors on the plane

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Cross product 1. The magnitude of the cross product is given by Joke presented on the television sitcom Head of the Class. The teacher asks: "What do you get when you cross an elephant and a grape?" The answer is "Elephant grape sine-theta." Whats the dot product ???

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Examples 1.Find an equation of the plane through the point (5,-2,4) with normal vector a=(1,2,3) 2.Prove that the planes and are parallel

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3. Find an equation of the plane which satisfies the stated conditions: 1) through P (-2,5,-8) with normal vector a=(-1,-4,1) 2) through P (2,5,-6) and parallel to the plane 3) through the points P(3,2,1),Q(-1,1,2),R(3,-4,1) 4. find the distance from the point P to the given plane 1) p=(2,1,-1), 2) p=(-2,5,-1),

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Angle ; projection of a line on a plane 1. Normal n projection L A B AB is the projection of line L on the plane The angle between line L and its projection on the plane 2. Angle between a line and the plane Projection :

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Angle between two planes 1. n1 n2 2. The angle between two planes is From dot product to find the angle theta First find the angle between two normal to the planes

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Find the common perpendicular 1*.cross product uxv is perpendicular to both u and v. For example: find the common perpendicular to u=(1,2,3) and v=(7,8,9)

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Find the common perpendicular vector 1.Technique introduced in the textbook. the common perpendicular vector of and is

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Find the normal to the plane Procedure: 1.Find the two vectors lie on the plane 2. find cross product of the two vectors Example: find the Cartesian equation of the plane through A(1,2,1), B(2,-1,-4) and C(1,0,-1) Two vectors on the plane: AB=(-1,3,5) and AC=(0,2,2) The vector is perpendicular to both AB and AC Therefore -4i+2j-2k is one of the normal to the plane,equation is -4x+2y-2z=-2

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5. If a line L has parametric equations find the plane contain L and point P=(5,0,2)

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Line A Line B Intersect Parallel (- identical?) Skew Line A Point B B is on the line A Find the perp. distance of the point from the line Line A Plane B Line is in the plane B A is parallel to B Line intersects the plane at one point Point A Plane B Find the perp. distance of the point from the plane A is in plane Angle between lines

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Plane A Plane B There is a line of Intersection m Parallel (- identical?) Angle between planes Given the two vector equations of the lines, you can examine the three simultaneous equations: If two equations yield a specific pair of values (t,s) and the third equation is consistent, then there is an intersection. If there are no solutions, the lines are skew or parallel. If q is a multiple of p, the lines are parallel, otherwise they are skew. If there are an infinite number of solutions, the lines are identical. θ The angle between two lines is easily found using the dot product. p q

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To examine whether a point is on a line, simply find a value of the Parameter t which satisfies the equation for each coordinate: If there is no value of t which makes these equations all true, then the point m is not on the line. To find the distance from the point m to the line: calculate the unit normal vector Find the displacement vector from A to m Take the dot product of these two vectors Use Pythagoras to find the third side of the triangle M A d X

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Line A Plane B Given a plane: and a line: … the line is parallel to the plane if: Moreover it is in the plane, also, if: This must be true for all t, so t must cancel out and the LHS=RHS= a constant. If there are no solutions for t:the line is parallel to the plane, but not in it. If there is one solution for t:the line intersects the plane. In the last case,

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Point A Plane B Given a plane: and a point: There are only two possibilities: the point is in the plane, or it isnt. To check if it is in the plane: simply substitute the coordinates of m into the equation of the plane. To find the distance of the point from the plane: n A (a,b,c) d m The simplest way is to find a unit normal vector and then calculate:

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Plane A Plane B Given a plane: and a plane: The planes may be parallel or identical or …? Intersect in a line n1 n2 This diagram shows how to find the angle between the planes.

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Plane A Plane B Given a plane: and a plane: To find the equation of the line: Choose any two values of x. Use the two plane equations to find the corresponding y and z which solves both plane equations Now you have two points on the line, so you can find its equation

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Finding a common perpendicular Read section 13.4 p carefully. The method described is actually the method to find the cross product. Find a vector perpendicular to each pair of vectors: Note that this method is a good way to find the equation of a plane through three points A, B, C:

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