makes angles °, °, ° with the x , y and z -axes respectively, find its direction cosines. . Find the direction cosines of a line which makes equal angles with the coordinate axes. .
If a line has the direction ratios – , , – , then what are its direction cosines ? . Show that the points ( , , ), (– , – , ), ( , , ) are collinear. .
Find the direction cosines of the sides of the triangle whose vertices are ( , , – ), (– , , ) and (– , – , – ). . 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 (i) it passes through a given point and has given direction, or (ii) it passes through two given points.
. . Equation of a line through a given point and parallel to given vector Let be the position vector of the given point A with respect to the origin O of the rectangular coordinate system. Let l be the line which passes through the point A and is parallel to a given vector .
Let be the position vector of an arbitrary point P on the line (Fig . ). Then AP is parallel to the vector , i.e., AP = λ , where λ is some real number. But AP = OP – OA i.e.
λ = Conversely, for each value of the parameter λ , this equation gives the position vector of a point P on the line. Hence, the vector equation of the line is given by » r a + Remark If ai bj ck , then a , b , c are direction ratios of the line and conversely, if a , b , c are direction ratios of a line, then ai bj ck will be the parallel to the line. Here, b should not be confused with | |. Derivation of cartesian form from vector