Cartesian equations of the required line are Here, ( , ( , , ) x y z and direction ratios of the required line are proportional to , , − . Therefore, Cartesian equations of the straight line are Example . The vector equation in parametric form of a line is ( ) ( ) . Find (i) the direction cosines of the straight line (ii) vector equation in non-parametric form of the line (iii) Cartesian equations of the line.
Comparing the given equation with equation of a straight line r tb , we have . Therefore, (i) If ˆ b i b j b k , then direction ratios of the straight line are b b b . Therefore, direction ratios of the given straight line are proportional to , , , and hence the direction cosines of the given straight line are (ii) vector equation of the straight line in non-parametric form is given by ( = . Therefore, ( )) ( ) (iii) Here ( , ( , , ) x y z and the direction ratios are proportional to , , Therefore, Cartesian equations of the straight line are Vector - - Example .
Find the vector equation in parametric form and Cartesian equations of the line passing through ( , , ) and is parallel to the line −− Rewriting the given equations as / and comparing with we have ( ) b i b j b k . Clearly, b is parallel to the vector . Therefore, a vector equation of the required straight line passing through the given point ( , , ) and parallel to the vector in parametric form is r = ( ) ( ), ∈ . Therefore, Cartesian equations of the required straight line are given by x + Example .
Find the vector equation in parametric form and Cartesian equations of a straight passing through the points ( , , ) and ( , , ) . Find the point where the straight line crosses the xy