Three Dimensional Co-ordinate Geometry

Three Dimensional Co-ordinate Geometry - Three Dimensional...

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Three Dimensional Co-ordinate Geometry Advanced Level Pure Mathematics Advanced Level Pure Mathematics Calculus II Chapter 7 Vectors 7.8 Vector Equation of a Straight Line 2 Chapter 10 Three Dimensional Coordinates Geometry 10.1 Basic Formulas 6 10.2 Equations of Straight Lines 6 10.3 Plane and Equation of a Plane 14 10.4 Coplanar Lines and Skew Lines 28 7.8 Vector Equation of a Straight Line c t a r + = , parameter scalar : t line straight on the point fixed a of ector position v : a vector direction : c Prepared by K. F. Ngai Page 1 10
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Three Dimensional Co-ordinate Geometry Advanced Level Pure Mathematics line straight on point any of ector position v : r Remark ) ( a b t a r - + = or b t a t r + - = ) 1 ( Example Find the vector equation of the straight line 1 , in the direction of k j i 2 2 + - and passing through the point with position vector ) 3 , 2 , 1 ( - - . Solution Example Find the vector equation of the straight line through the point ) 4 , 5 , 3 ( - in the direction of k j i + - . Find also the point on this line which has i 4 as one component vector of its position vector. Solution Example Find the equation of the line joining the points ) 6 , 2 , 1 ( - - A and ) 3 , 8 , 4 ( - B . Find the coordinates of the point of intersection of this line and the x-y plane and the ratio in which x-y plane divides AB . Solution Prepared by K. F. Ngai Page 2
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Three Dimensional Co-ordinate Geometry Advanced Level Pure Mathematics Example Let ) 0 , 7 , 8 ( - = A and ) 3 , 1 , 2 ( - - = B . (a) Find the equation of the straight line AB . (b) Find the perpendicular distance from the point ) 9 , 7 , 4 ( - P to the line AB . Find also the foot of perpendicular. Solution Example The line joining two points ) 1 , 8 , 1 ( - P and ) 2 , 4 , 4 ( - Q meets the - xz and - yz planes respectively at R and S . Find the coordinates of R and S and the ratios in which they divide PQ . Solution Prepared by K. F. Ngai Page 3
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Three Dimensional Co-ordinate Geometry Advanced Level Pure Mathematics Remark In above example (b), the distance from P to AB may also be found directly without calculating the foot of perpendicular. The method is outlined as follows: By referring to Figure, AB AP AB AP PR × = = θ sin Since Example By finding the foot of perpendicular from the point ) 13 , 1 , 10 ( - P to the line, ) 5 (4 5 : j i t k i r L - + + = , find the equation of straight line passing through P and perpendicular to L , find the perpendicular distance from P to L . Solution Prepared by K. F. Ngai Page 4
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Three Dimensional Co-ordinate Geometry Advanced Level Pure Mathematics Three Dimensional Co-ordinate Geometry 10.1 Basic Formula The Distance Between Two Points Distance between ) , , ( 1 1 1 z y x A and ) , , ( 2 2 2 z y x B is 2 2 1 2 2 1 2 2 1 ) ( ) ( ) ( z z y y x x - + - + - . Prepared by K. F. Ngai Page 5
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Three Dimensional Co-ordinate Geometry Advanced Level Pure Mathematics Section Formula Let ) , , ( z y x P divide the joint of ) , , ( 1 1 1 z y x A and ) , , ( 2 2 2 z y x B in the ratio n m PB AP = The Coordinate of the point P is + + + + + + n m nz mz n m ny my n m nx mx 1 2 1 2 1 2 , , 10.2 Equations of Straight Lines In vector form, the equation of straight line is c t a r + = , where r
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