( vi ) The projection of the line segment joining points P ( x1 , y1 , z1 ) and Q ( x2 , y2 , z2 ) to the line having direction cosines 1 , m , n is |( x2 – x1 ) l + ( y2 – y1 ) m + ( z2 – z1 ) n |.
( vii ) The direction ratio of the line passing through points P ( x1 , y1 , z1 ) and Q ( x2 , y2 , z2 ) are proportional to x2 – x1 , y2 – y1 – z2 – z1 Then , direction cosines of PQ are
x2 – x1 / | PQ |, y2 – y1 / | PQ |, z2 – z1 / | PQ |
Area of Triangle
If the vertices of a triangle be A ( x1 , y1 , z1 ) and B ( x2 , y2 , z2 ) and C ( x3 , y3 , z3 ), then
Angle Between Two Intersecting Lines
If l ( x1 , m1 , n1 ) and l ( x2 , m2 , n2 ) be the direction cosines of two given lines , then the angle θ between them is given by
cos θ = l112 + m1m2 + n1n2 ( i ) The angle between any two diagonals of a cube is cos -1 ( 1 / 3 ).
( ii ) The angle between a diagonal of a cube and the diagonal of a face ( of the cube is cos -1 ( √2 / 3 )
Straight Line in Space
The two equations of the line ax + by + cz + d = 0 and a ’ x + b ’ y + c ’ z + d ’ = 0 together represents a straight line .
1 . Equation of a straight line passing through a fixed point A ( x1 , y1 , z1 ) and having direction ratios a , b , c is given by