Three Dimensional Geometry. Linear Programming

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Three Dimensional Geometry Linear Programming A plane is determined uniquely if any one of the following is known: The normal to the plane and its distance from the origin is given, i.e. equation of a plane in normal form.

If the planes are at right angles, then θ = 90 0 and so cos θ = 0. Hence, cos θ = A 1 A 2 + B 1 B 2 + C 1 C 2 = 0. If the planes are parallel then.

1 Three Dimensional Geometry Linear Programming A plane is determined uniquely if any one of the following is known: The normal to the plane and its distance from the origin is given, i.e. equation of a plane in normal form. It passes through a point and is perpendicular to a given direction, It passes through three given non collinear points. Now we shall find vector and Cartesian equation of the planes. If the planes are at right angles, then θ = 90 0 and so cos θ = 0. Hence, cos θ = A 1 A 2 + B 1 B 2 + C 1 C 2 = 0. If the planes are parallel then. If the equation of the plane π2 is in the form, where is normal to the plane, when the perpendicular distance is. The length of the perpendicular form origin O to the plane. = d is (since = 0).

2 Direction cosines of a line are the cosines of the angles made by the line with the positive directions of the coordinate axes. If l, m, n are the direction cosines of a line, then l2 + m2+ n2 = 1. Direction cosines of a line joining two points P(x 1, y 1 z 1,) and Q (x 2, y 2, z 2 ) are Where PQ = Direction ratios of a line are the numbers which are proportional to the direction cosines of a line. If l, m, n are the direction cosines an a, b, c are the direction ratios of a line then Skew lines are lines in space which are neither parallel nor intersecting they lie in different planes. Angle between skew lines is the angle between tow intersecting lines drawn from any point (preferably through the origin) parallel to each of the skew lines. If l 1, m 1, n 1, and l 2, m 2, n 2, are the direction cosines of two lines; and θ is the acute angle between the two lines; then Cos θ = Vector equation of a line that passes through the given point whose position vector is and parallel to a given vector. Equation of a line through a point (x 1, y 1, z 1,) and having direction cosines l, m, n, is

3 The vector equation of a line which passes through two points whose position vectors are Cartesian equation of a line that passes through two points (x 1, y 1, z 1 ) and (x 2, y 2, z 2,) is. If θ is the acute angle between then cosθ = If are the equations of two lines, then the acute angle between the two lines is given by cos θ = l 1 l 2 + m 1 m 2 + n 1 n 2 Shortest distance between tow skew lines is the lines segment perpendicular to both the lines. Shortest distance between is Shortest distance between the lines: and is Distance between parallel lines In the vector form, equation of a plan which is at a distance d from the origin, and is the unit vector normal to the plane through the origin is Equation of a plane which is at a distance of d from the origin and the direction cosines of the normal to the plane as l, m, n, is lx + my + nz = d. The equation of a plane through a point whose position vector is and perpendicular to the vector is ( ). = 0.

4 Equation of a plane perpendicular to a given line with direction ratios A, B, C and passing through a given point (x 1, y 1, z 1,) is A(x x 1 ) + B(y y 1 ) Type equation here.+ C(z z 1 ) = 0 Equation of a plane passing through three non collinear points (x 1, y 1, z 1,) (x 2, y 2, z 2,) and (x 3, y 3, z 3,) is Vector equation of a plane that contains three non collinear points having position vectors and is Equation of a plane that cuts the coordinates axes at (a, 0, 0), (0, b, 0) and (0, 0, c) is Vector equation of a plane that passes through the intersection of planes constant. where λ is nay nonzero Cartesian equation of a plane that passes through the intersection of two given planes A 1 x + B 1 y + C 1 z + D 1 = 0 and A 2 x + B 2 y + C 2 z + D 2 = 0 is (A 1 x + B 1 y + C 1 z + D 1 ) + λ(a 2 x + B 2 y + C 2 z + D 2 ) = 0. Two lines In the Cartesian form above lines passing through the points A(x 1, y 1, z 1,) and B(x 2, y 2, z 2 ) = In the vector form if θ is the angle between the two planes,

5 The angle φ between the line and the plane is sin φ = The angle θ between the planes A 1 x + B 1 y + C 1 z + D 1 = 0 and A 2 x + B 2 y + C 2 z + D 2 = 0 is given by cos θ = The distance of a point whose position vector is from the plane The distance from a point (x1, y1, z1) to the plane Ax + By + Cz + D = 0 is Linear Programming A linear programming problem is on e that is concerned with finding the optimal value (maximum or minimum) of a linear function of several variables (called objective function) subject to the condition that the variables are non negative and satisfy a set of linear inequalities (called linear constraints). Variables are sometimes called decision variables and are non-negative. A few important linear programming problems are: (i) (ii) (iii) Diet problem Manufacturing problems Transportation problems

6 The common region determined by all the constraints including the non-negative constraints x 0, y 0 of a linear programming problem is called the feasible region (or solution region) for the problem. Points within and on the boundary of the feasible region represent feasible solutions of the constraints. Any point outside the feasible region is an infeasible solution. Any point in the feasible region that gives the optimal value (maximum or minimum) of the objective function is called an optimal solution. The following theorems are fundamental in solving linear programming problems; Theorem 1 Let r be the feasible region (convex polygon) for a linear programming problem and let Z = ax + by be the objective function. When Z has an optimal value (maximum or minimum), where the variables x and y are subject to constraints described by linear inequalities, this optimal value must occur at a corner point (vertex) of the feasible region. Theorem 2 Let R by the feasible region for a linear programming problem, and let Z = ax + by be the objective function. If R is bounded, then the objective function Z has both a maximum and a minimum value on R and each of these occur at a corner point (vertex) of R. If the feasible region is unbounded, then a maximum or a minimum may not exist. However, if it exists, it must occur at a corner point of R.

7 Corner point method for solving a linear programming problem. The method comprises of the following steps: Find the feasible region of the linear programming problem an determine its corner points (vertices). Evaluate the objective function Z = ax + by at each corner point. Let M and m respectively be the largest and smallest values at these points. If the feasible region is bounded, M an m respectively are the maximum and minimum values of the objective function. If the feasible region is unbounded then, M is the maximum value of the objective function, if the open half plane determined by ax + by > M has no point in common with the feasible region. Otherwise, the objective function has no maximum value. m is the minimum value of the objective function, if the open half plane determined by ax + by < m has no point in common with the feasible region. Otherwise, the objective function has no minimum value. THANK YOU

THREE DIMENSIONAL GEOMETRY

For more important questions visit : www4onocom CHAPTER 11 THREE DIMENSIONAL GEOMETRY POINTS TO REMEMBER Distance between points P(x 1 ) and Q(x, y, z ) is PQ x x y y z z 1 1 1 (i) The coordinates of point