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collinear Solution Direction ratios of line joining A and B are 1 2 2 3 3 4ie 1 5 7 The direction ratios of line joining B and Care 3 1 8 2 11 3 ie 2 10 14 It is clear that direction ratios of AB and BC are proportional hence AB is parallel to BC But point B is common to both AB and BC Therefore A collinear points B C are EXERCISE 11 1 1 Ifa line makes angles 90 135 45 with the x y axes respectively find its direction cosines 2 Find the direction cosines of a line which makes equal angles with the coordinate axes 3 If a line has the direction ratios 18 12 4 then what are its direction cosines 4 Show that the points 2 3 4 1 2 1 5 8 7 are collinear 5 Find the direction cosines of the sides of the triangle whose vertices are 3 5 4 1 1 2 and 5 5 2 11 3 Equation of a Line in Space We have studied equation of lines in two dimensions in Class XI we shall now study the vector and cartesian equations of a line in space A line is uniquely determined if 1 it passes through a given point and has given direction or i it passes through two given points 11 3 1 Equation of a line through a given point and parallel to a given vector h Let a be the position vector of the given point A with respect to the origin O of the rectangular coordinate system Let be the line which passes through the point A and is parallel to a given vector b Let 7 be the position vector of an arbitrary point P on the line Fig 11 3 382 MATHEMATICS Then AP is parallel to the vector b i e AP b where is some real number But i e Rationalised 2023 24 AP OP 0A 2b 7 a Conversely for each value of the parameter A this equation gives the position vector of a point P on the line Hence the vector equation of the line is given by F a b and a ishd A P Fig 11 3 Pl Y 1 Remark If h ai b ck then a b c are direction ratios of the line and conversely if a b c are direction ratios of a line then b ai bj ck will parallel to the line Here b should not be confused with bl Derivation of cartesian form from vector form Let the coordinates of the given point A be x y z the direction ratios of the line be a b Consider the ates of any point Pbe x y z Then x i y j z k 7 xi y zk b al b ck Substituting these values in 1 and equating the coefficients of 1 and k we get x x a y y b z z kc 2 These are parametric cautions of the line Fliminating the parameter from 2

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