(3) the Cartesian equations of the line.3 ways of expressing the line in space (1) the vector equation of the line, (2) the parametric equations of the line,

(1) The vector equation of the straight line

xyztaRALvLine L is parallel to v.

and v = ai + bj + ck is a direction vector of the lineSuppose Position vector R that is OR = (x , y , z) is a point which is free to move on a line ,Position vector A is given as OA= (x1, y1, z1)

Isolating t in each of these equations gives = = 3) CARTESIAN EQUATIONNoteis the numerator and that the components of is the denominatorThe coordinates of point

In finding the vector equation of straight line1. A given point on the line2. A vector parallel to the line

A line L passes through point A (1,-4, 2) and is parallel to Find (a) the vector equation,(b) the parametric equations,(c) the Cartesian equations for line L.

Angle between two straight linesFor two straight lines,The angle between and

is

2 angles obtained. That is both acute angle and obtuse anglev1v2

The angle between 2 lines

The angle between 2 linesThe two lines have the equations r = a + tb and r = c + sd. The angle between the lines is found by working out the dot product of b and d. We have b.d = |b||d| cos A.

ExampleFind the acute angle between the lines

Direction Vector of L1, b1 = 2i j + 2k Direction Vector of L2, b2 = 3i -6j + 2kIf is the angle between the lines,Cos =

Example Cos = Cos = = 40 22